3 a 1* (D Al su T (D 1 physics for teachers a modern review Weber I WIGAN PUBLIC LIBRARIES ■ ?A crn ;qp7 Physics for Teachers: A Modern Review Physics for Teachers: A Modern Review Robert L. Weber Associate Professor of Physics The Pennsylvania State University \ vV' McGraw-Hill Book Company New York Son Francisco Toronto London to Marion, Robert, Karen, Meredith, and Ruth who were patient -» PHYSICS FOR WIQAN UBRARfES WITH DRAWN FOR EW Copyright © 1964 by McGraw-Hill, Inc. All Rights Reserved. Printed in the United States of America, This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number 63-19320 68806 Preface The reader of this boob is assumed to be interested in space physics and atomic physics and to have had a full-year course in general physics at the college level. Capitalizing on his interest in rockets and satellites. Part 1 presents enough of the principles of mechanics and electricity to serve as a good basis for under- standing nineteenth-century classical physics. In Part 2, with the atom as the central theme, the theories of relativity and quantum physics which have characterized twentieth-century modern physics are developed. In both parts important topics such as wave properties and relativity, which are likely to be less familiar to the reader, are developed in greater detail. Some calculus notation is used,- but where feasible (e.g., in Sec. 4.4), noncalculus explanations are used, and several derivations in- volving integration are subordinated in an Appendix. An aim of this book is to encourage the reader, whether a student or a mature teacher, to appreciate the relatedness of the various fields of science and to be willing to venture into new areas with the ability he has gained from intensive study of a few selected areas. In the planning of this book I am indebted to the interest of students and colleagues in several science institutes sponsored by the National Science Foundation. For six years I served as vii Preface associate director, director, and teacher of the physics part of programs at The Pennsylvania State University; Colorado State University, and Yale University. The present text evolved from the study guide used in the last-mentioned program. I express indebtedness to that scholarly textbook "Physics for Students of Science and Engineering," by David Halliday and Robert Resniek, for the manner of presentation used in the first part of Chap. 13. Contents Robert L. Weber Preface vll Part 1 Looking Out: Rockets, Satellites, Space Travel 1 What's Up? 3 Environment of Space 9 Rocket Propulsion 20 Escape from Earth 40 Satellites 50 Motion of Bodies in Space 60 Travel to Moon and Planets 68 Part 2 Looking In: Atomic and Nuclear Physics 75 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. The Atomic Idea 77 Wave Motion 91 Electric and Magnetic Forces 116 The Electron 136 Ions and Isotopes 149 Electromagnetic Radiation I Classical Theory 156 II Quantum Effects 167 Relativity Wonderland 184 Hydrogen Atom Bohr Model 203 Quantum Dynamics 216 Radioactivity 238 Nuclear Reactions 252 Absorption of Radiation 260 Unconventional Energy Sources 272 Appendices A. Reaction Thrust 289 B. Burnout Velocity and Range 291 C. Schrodinger Wave Equation 293 D. 1. Books for a Physics Teacher's Reference Shelf 295 2. Some Periodicals for a School Science Library 296 3. Professional Organizations of Interest to the Physics Teacher 296 4. Some Suppliers of Physics Apparatus for Teaching 297 5. Greek Alphabet 297 6. Symbols 298 7. Instruction on the Use of a Slide Rule 298 7A. Slide Rule Bibliography 300 8. Trigonometric Functions 301 9. Logarithms to the Base e 302 10. Values of Physical Constants 304 11. Conversion of Electrical Units 305 Index 307 PART Looking Out: Rockets, Satellites, Space Travel What's Up? In his millennia of looking at the stars, man has never found so exciting a challenge as the year 1957 has sud- denly thrust upon him. Fred L. Whipple 1.1 ASTRONAUTICS TODAY Fictional accounts of space travel had been written before the lime of Jules Verne. In the second century a.i>., Lucian of Santos wrote of a visit to the moon. But the foundation for converting fantasy into an engineering possibility was the invention of lite rocket. Most current progress in the science and technology of space flight is an outgrowth of the efforts since World War II to develop long-range military missiles. Popular concern aboul space technology was aroused when the first artificial earth satellite was launched by the Soviet Union in 1!J">7. In the next 3 years, some 36 satellites carrying instru- ments were launched, 30 of them by the United States. Also dur- ing this period several moon shots and space probes sought infor- mation farther from the earth. Next came the spectacular manned orbital flights made l>y astronauts of the Soviet Union and the United States. 3 4 Looking Out: Rockets, Satellites, Space Travel In the "World Almanac" one may find a listing of the major space vehicles launched since 19">7. Table 1.1 gives the longer perapective of the history of man's use of rockets. Table 1.1 c. 300 c. 1200 c. 1780 1792 c. 1800 1812 c 1830 1846 1913 C 1915 1926 c 1930 1931 1932 1936 1941 1942 1944 c . 1945 1946 1949 c . 1952 Milestones in missilery Hero of Alexandria uses the reacting force of escaping steam to propel an experimental device. Chinese use gunpowder to propel "arrows of flying fire," equiva- lent to present-day skyrockets. Advanced type of rocket developed in India. Troops of Tipu, Sultan of Mysore, use rockets against British in second Mysore War. Sir William Congreve of Great Britain improves rocket propellant to provide considerable increase in range. British use rockets in attack on Fort McHenry (Baltimore), com- memorated in the line "... and the rockets' red glare" in our National Anthem. William Hale, an American, increases stability of rockets by adding nozzle vanes. Mexican War sees first use of rocket weapons by United States in o war. Lifesaving rockets developed by English and German inventors. Ramjet proposed and patented in France. World War I sees advent of guided missile to supplant aimed rockets. Dr. Robert H. Goddard, professor of physics at Clark University, fires first successful liquid- fuel rocket. Germans experiment with the pulse jet, used to power the Nazi V-1 "bun bomb" of World War II. Germany uses liquid rocket fuel. Captain Walter Dornberger undertakes development of liquid- fuel rocket weapons for the German Army. German Peenemunde Project is organized, to develop war rockets. United States starts work on controllable rocket weapons. American Razon missile, controllable in both azimuth and range, is developed. United States government awards first contract for research and development of guided missile to General Electric Company. Germany uses V-1 buzz bomb, V-2, and other rocket missiles in World War II. United States uses "Weary Willie" unmanned bombers. Work is started in the United States on an intercontinental ballistic missile program, the MX-774, First flight of a missile beyond earth's atmosphere is made at White Sands, N. Mex. United States long-range missile program is stimulated by Atomic Energy Commission warhead developments. What's Up? 5 Table 1,1 Milestones in missilery (continued) 1954 United States starts ICBM program; USAF awards contracts to Convair, North American Aviation, and General Electric. 1957 First artificial earth satellites — Sputniks I and II — launched by rocket (October 4 and November 3). 1.31.58 Explorer I, first United States satellite, launched. 3.17.58 Vanguard I, first "permanent" satellite launched by the United States. 10,11.58 Pioneer I, first lunar probe, launched by the United States. 12.18.58 Project Score (Atlas) launched, broadcasting a human voice from outer space for the first time. 1.2.59 Russia launches Lunik, first satellite to orbit around sun. 3.3.59 Pioneer IV launched, first United States satellite to orbit sun. 9.12.59 Russia launches first space vehicle to land on moon. 10.4.59 Russia launches first satellite to orbit moon. 8.11.60 United Slates recovers first space vehicle from orbit. 4.12.61 Manned orbital flight achieved in Soviet Vostok satellite, 2.20.62 Pfoject Mercury succeeds in manned orbital flight, 7.10.62 Telstar satellite relays first transatlantic television, programs. 8.13.62 Two Russian astronauts put in reloted orbits. 8.27.62 Mariner II launched to encounter Venus. 12,14.62 Mariner II passed within 22,000 mi of Venus, reporting data on temperature, cloud cover, magnetic field, particles and radiation dosage encountered throughout voyage. 1.2 ASTRONAUTICS TOMORROW What feat may be expected, perhaps in the next 2 or 3 years, from adaptations of the intercontinental ballistic missiles already available? The staff of the RAND Corporation has estimated that we shall be able to do the following: 1. Orhit satellite payloads of 10,000 lb to 300 mi altitude 2. Orbit satellite payloads of 2,500 lb at 22,000 mi 3. Impact 3,000 lb on the moon 4. Land, intact, more than 1,000 lb of instruments on the moon, Venus, or Mars ."). Probe the atmosphere of Jupiter with 1,000 lb of instruments fi. Place a man, or men, in a satellite orbit around the earth for recovery after a few days of flight 1.3 WHO SHOULD CARE? By the military, the costly development of the rocket has been pushed chiefly as a gunless artillery device and for bombardment 6 Looking Out: Rockets, Satellites, Space Travel over intercontinental distances with nuclear warheads. Rocket- launched viewing satellites may make possible the inspection of foreign territory and thus discourage preparation for war. The difficulties that statesmen now have in reaching agreement on inspection and on disarmament will, however, probably increase as military space technology expands. In addition to the military reasons, there are many scientific incentives for making satellites. Some important problems await- ing investigation are: 1. Determination of density, pressure, and temperature in the upper layers of the atmosphere 2. Exact measurement of the dimensions of the earth, the conti- nental distances, and other geodetic measurements :i. A detailed study of radiation from the sun 4. Observation of the intensity of cosmic rays and other radia- tion in the earth's atmosphere f). Correlation of the currents of nuclei, neutrons, and other particles flying toward the earth with sunspot activity (i. Kstimation of the distribution of mass in the earth's crust from the orbital planes of the artificial earth satellite 7. Study of the propagation of radio waxes in the upper atmos- phere and provision of radio communication, navigation bea- cons, and television with the aid of satellites 8. Improvement in weather forecasting it. Making feasible astro no mica I investigations without atmos- pheric and other disturbances 10. Study of biological specimens in environments different from that on earth Although we have these incentives for space exploration, it is likely that such exploration will enlighten us in fields even beyond our present speculations. Space flight obviously demands development of devices of great reliability 1o operate for long periods under extreme condi- tions of environment. Engineering advances depend on funda- mental scientific knowledge, and in the past these advances have contributed tools for the obtaining of new knowledge. This inter- action or feedback is occurring in astronautics. When spaceships can carry instruments, or man himself, into other parts of the solar system, new information will surely become available for the physicist, biologist, and astronomer. What's Up? 7 The philosopher and the theologian are already adapting their thinking to the eventuality that man may encounter life else- where in the universe. It seems probable that just as the tele- scope profoundly altered seventeenth-century thought, the space vehicle will extend twentieth-century man's understanding of the universe and his role in it. 1.4 CAREERS IN ASTRONAUTICS Astronautics touches almost all fields of current science and tech- nology. 1 1 may be expected to lead to entirely new fields. Entrance into the field of astronautics can be made by one who has acquired knowledge in one or more of these fields: mechanical, aeronaut- ical, and electrical engineering; mathematics; physics; biophys- ics; and chemistry. Mathematics and physics are basic. With nuclear power a necessity for distant space travel, the field of nuclear physics is of special importance. Information about careers in astronautics can be obtained from the corporations active in this field, and also from the agency which coordinates the government's activities, NASA. The .Na- tional Aeronautics and Space Administration was created by an act of Congress signed by President Eisenhower on July 2H, I9">8. The act declared that "it is the policy of the United States that activities in space should be devoted to peaceful purposes for the benefit of mankind." FILMS ON ASTRONAUTICS For a listing of some 9(1 films on rockets, missiles, and space science see It. L Weber: Films for Students of Physics, Supplement I, American Jcwntd of I'h units, SO: 321 327 (19«2). SUGGESTIONS FOR FURTHER READING IN ASTRONAUTICS Hooks: Adams, Carsbie C: "Space Flight, " McGraw-Hill Book Company, Inc., New York, I95S, 373 pp. Hcnson, t). 0., and H. Strugliold: "Physics and Medicine of the Atmos- phere ami Space," .John Wiley A Sons, Inc., New Yolk. IWiO, •»4f) pp. "uchheim. It. W„ and Staff of HAND Corp.: "Space Handbook," House Document 80, U.S. Government Priming Office, Washing- 8 Looking Out: Rockets, Satellites, Space Travel ton 25, D.C, 1959. Also Random House, hit-. (Modern Library edition), New York. The National Aeronautic* and Space Administration: "Space: The New Frontier," U.S. Government Printing Office, Washington 25, D.C., 1962,48 pp. Ordway, Frederick I.: "Annotated Bibliography of Spaee Science and Technology," Arfbf Publications, P.O. Box 6285, Washing! on 15, D.C., 1011-2. Scifert, Howard S. (ed.): "Sparc Technology," .John Wiley & Sons, Int.. New York, 1959. Text based on graduate-level lectures presented by University Extension, University of California, in cooperation with Hamo-Wooldridgc Corp. Periodicals: Astronautics. Published monthly by The American Pocket Society, Inc., 500 Fifth Ave, New York W. NY. Aviation HVefc (Including Space Technology). McGraw-Hill Publishing Company, Inc., 330 West 42 St., New York 36, N.Y. Missiles and Rockets. Published weekly by American Aviation Publi- cations, 1001 Vermont Ave., NW, Washington 5, D.C. Sky and Telescope. Published monthly by Sky Publishing Co., 60 Garden St., Cambridge 38, Mass. Space Aeronautics. Published monthly by Conover-Mast Publications. foe., 20.") East 42 St., New York 1 7, N.Y. Space Age. Published quarterly bv Quinn Publishing Co., Kingston, N.Y. Spaceflight. Published bimonthly hy British Interplanetary Society, 12 Bessborough Gardens, London, SW 1. England. Space Journal. Published quarterly by Space Enterprises, Inc., P. O. Box 94, Nashville, Tenn. Publications of Sperial Interest to Students: Adams. Carsbie C, Wernher von Braun, and Frederick I. Ordway: "Careers in Astronautics ami Rocketry," McGraw-Hill Book Com- pany, Inc., New York. 1902, 248 pp. Map of the Moon, chart, 45 by 35 in.. General Electric Missile and Space Vehicle Department, Valley Forge Space Technology Cen- ter (Mail: P.O. Box 8555, Philadelphia 1. Pa.). Map of Outer Spaee, chart, 28 by 25 in., General Electric Missile and Space Vehicle Department. "Short Glossary of Space Terms," National Aeronautics and Spaee Administration, U.S. Government Printing Office, Washington 25, D.C, 1962, 57 pp. "Space Primer: An Introduction to Astronautics," Convair-Asfro- nautics. Dcpl. 120, P.O. Box 112s, San Diego 12, Calif., 72 pp. Environment of Space The most incomprehensible thing about the universe is that it is compre- hensible. Albert Einstein 2.1 INTRODUCTION Our sun, the 9 major planets, 31 known moons, and thousands of lesser bodies all revolving around the sun constitute the solar -v-ii'iii. The planets move around the sun in the same direction in elliptical orbits which are nearly circular (big. 2.1). All the orbits lie in nearly the ecliptic plane of the sun's apparent path among the stars. The orbit of Pluto deviates most, about 17°. from the ecliptic plane. The zone about 17° wide on each side of the ecliptic plane is known as the zodiac. The average distance of the earth from the sun is 92,900,000 mi, a distance which is defined as one astronomical unit (a.u.). The diameter of the orbit of Pluto, the outermost member of the solar system, is about 79 a.u. The four inner planets, Mercury, Venus, Earth, and Mars, are sometimes called the terrestrial planets, They are relatively 9 10 Looking Out: Rockets, Satellites, Space Travel small, dense bodies. The next four outer planets, Jupiter, Saturn, Uranus, and Neptune, are called the major planets or the giant planets. They are relatively large bodies with ice and rock cores % >. Meteors Jupiter Fig. 2.1 Solar system; orbits of Mercury, Venus, Eorth, Mars, and Jupiter. below their visible atmospheres. Physical data on objects of prin- cipal interest in the solar system are given in Table 2.1. Table 2.1 Physical data on some bodies in the solar system Body Mean Man, Diameter, Gravita- Intensity Length Length distance times mi tional force of of day of year from earth's at solid sunlight, tun, mass surface. rel. to a.u. 9'« earth Sun 329,000 864,000 - Mercury 0.39 0.05 3,100 0.3 6.7 68 d 88 d Venus 0.72 0.82 7,500 0.91 1.9 ? 225 d Earth 1 1 7,920 1 1 24 hr 365 d Mars 1.52 0.11 4,150 0.38 0.43 24.6 hr 1.9 yr Jupiter 5.2 317 87,000 t 0.037 10 hr 12 yr Saturn 9.5 95 71,500 t 0.01 1 10 hr 29 yr Uranus 19.2 15 32,000 t 0.0027 11 hr 84 yr Neptune 30 17 31,000 t 0.0011 16 hr 165 yr Pluto 79 0.8 ? | 0.0006 ? 248 yr Moon 1.0 0.012 2,160 0.17 1 27 d * Hoi no solid surface. f location of solid surfoce not known (far below dense atmospheric gases). 2.2 THE SUN The sun, whose gravitational attraction chiefly controls the mo- tion of planets in the solar system, is classified as about average Environment of Space 11 among stars in size, in temperature, and in brightness (spectral type C-2). Its nearness to the earth makes the sun appear to us very large and bright. The surface temperature of the sun has Keen measured as about ti000°C, or I0,000°r', The energy output of the sun as light and heat is remarkably constant. Solar energy arrives at the surface of the earth at an average rate of 1.35 kilo- watts/ m-. This solar energy, resulting from a series of thermo- nuclear reactions, makes life possible on the earth. Sunlight takes a little more than 8 min to reach the earth. When analyzed with a spectrograph, sunlight is found to consist of a continuous spectrum, but with the colors crossed by many dark lines. The absorption lines are produced by gaseous materials in the atmosphere of the sun. From their lines, some 70 of the chemical elements occurring on earth have been identified as present in the sun. The radiating surface of the sun is called the photosphere. Above it is the chromosphere, visible to the unaided eye al times of total eclipses as a turbulent pinkish-violet layer. The pearly light of corona extends millions of miles beyond the chromo- sphere. Corona are related to the appearance of sunspots — dark, irregular regions which may extend several hundred thousand miles across and whicii may last for a few weeks to several months. The output of ultraviolet radiation, radio waves ("static"), and charged particles (cosmic rays) from the sun is highly variable. 2.3 THE PLANETS because of its nearness to the sun, Mercury is difficult to observe ami knowledge of its physical characteristics is not very accurate. Mercury has a mass about one-twentieth the mass of the earth. It lias no moon. .Mercury has a rockysurl'ace. probably similarlof hat »f our moon. Mercury always keeps the same side turned toward the sun. This side probably has surface temperatures as high as b)()°C, while the side in darkness is cold enough to retain frozen gases, with temperatures approaching absolute zero. Venus is slightly smaller than the earth, shrouded in a dense ; ilinospherc opaque to light of all wavelengths, Neither free oxy- Ren nor water vapor has been detected on Venus. Carbon dioxide ] s abundant in its atmosphere, with nitrogen and argon also 12 Looking Out: Rockets, Satellites, Space Travel present. It is thought that die surface of Venus is hot {about 425°C), dry, and dark beneath a continuous dust storm. Mars lias u diameter ahoul half that of the earth, its rate of revolution is about the same as that of earth, and its axis, too, is inclined about 2;">° from the plane of its orbit. Mars takes <»87 of our days to make one circuit of the sun. Although the orbit is nearly circular, it is not centered on the sun; Mars is more than 30 million mi farther from the sun at some parts of its year than at others. More than half of the surface of Mars is a desert of rusty rock, sand, and soil. The rest of the sin-face shows seasonal color changes which have been interpreted as due to vegetation. While noon summer temperature on Mars probably reaches 30°C, night temperatures probably fall to — 70°C. The atmosphere {mostly nitrogen) on Mars has a pressure about 10 per cent of the earth's atmosphere. Oxygen has not been detected. The white polar caps are probably frost layers, which on melting furnish moisture for the summer growth of vegetation. Mars has two small satellites about 5 and 10 mi in diameter. Each of the four giant planets, Jupiter, Saturn, Uranus, and Neptune, seems to have a dense rocky core surrounded by a thick layer of ice and covered by thousands of miles of compressed hydrogen and helium with smaller quantities of methane and ammonia. These planets receive relatively feeble radiation from the sun, so that the temperatures of their upper atmospheres range from -100 to -200°C. These planets rotate rapidly and in the same direction. Some of the satellites of these planets are larger than the earth's moon and may have physical character- istics less formidable for space-flight visits than the major planets themselves. Jupiter has 12 satellites. Four are bright enough to be easily visible with binoculars, and their rapid motion causes interesting changes in position from night to night. Saturn is the farthest of the planets visible to the unaided eye. It has nine satellites. Saturn is surrounded by remarkable flat rings in the plane of the equator. It has been suggested that the rings are made up of tiny particles of a shattered tenth satellite. 2.4 THE MOON The moon has a mass about B V that of the earth, a diameter of about 2,l(iO mi, and an elliptic orbit which gives it an average Environment of Space 13 distance from the earth of 239,000 mi. The moon has no appre- ciable atmosphere. Its surface, comprising many craters and high mountains, is probably dry, dust-covered rack. The moon rotates on its axis in a period of time equal to the period of its revolution about the earth, 27.3 days. The moon's elliptic orbit and its variation in altitude from season to season permit us to examine about 00 per cent of its surface, over a period of time. 2.5 ASTEROIDS, COMETS, AND METEORITES Asteroids are pieces of planetlike material which, unlike the planets, are of irregular shape. They may be the shattered frag- Fig. 2.2 Radiation pressure which forces the tail of a comet away from the sun might propel a spaceship. * — V meats of one or more planets. Most measure a few miles across; the largest, Ceres, is nearly f>00 mi across. The orbits of most asieroids lie between the orbits of Mars am! Jupiter {Fig. 2.1). The time for one revolution varies greatly among the asteroids. Comets are large, loose collections of material that penetrate the inner regions of the solar system from outer space. The most famous one, named after Halley, has been sighted every 7(i.02 years since 240 h.c, but not all return periodically. Comets have a head and tail. The head is made up of heavy particles and is attracted by the sun. The tail is made up of dust and gas and is forced away from the sun by radiation pressure as the comet sweeps past the sun. The brightness of the comets is probably due to reflected sunlight. The earth has passed through the tails of many comets without effect. Some 2,000 tons of material from outer space reaches the Ruth's atmosphere each day in the form of meteorite particles, 1'hose enter the earth's atmosphere with speeds of 10 to 50 mi/sec and are heated to incandescence, producing the light streaks called incteors. Reflection of radio waves from the ionized paths 14 Looking Out: Rockets, Satellites, Space Travel and observations of sky gk)W at twilight as well as direct visual counting of meteors indicate tliat a large amount of material is received daily, but data are inadequate. How much meteoi- itie material a space vehicle might encounter is an important unknown. In the night sky a faint tapered band of zodiacal light can |„, ..en and traced photoelectrically. It is evidence of cosmic dust, mierometeorites, concentrated toward the plane of the solar system. 2.6 RADIATION Beyond the shelter of the earth's atmosphere, x rays, ultravinl.t rays, and cosmic rays exist at intensities which may have to be considered in planning exploration by space vehicles. The WA Explorer satellite detected an encircling belt of high-energy radMr tion extending upward from a height of a few hundred miles, most intense in the equatorial region (Fig. 2.3). The earth's magnetic field traps the particles, chiefly electrons and protons, constituting the radiation belt. During solar flares, the sun delivers as much as 1,000 times its normal radiation. The nature of the radiation found in space is described in Table 2.2. In order to avoid subjecting astronaut* to radiation Table 2.2 Radiation in space Name Nature of Charge Mass radiation Photon Electronic gnetic Quantum Electronic gnetic Xray Electromagnetic Gamma ray Electromagnetic Electron Particle — a lm, Proton Particle +■ 1,840m, or 1 omu Neutron Porticle 1,841m, Alpha porticle Particle + 2. 4 amu Heovy primaries Porticle &+3e ' 6 amu Where found Radiation belts, solar radia- tion (produced by nuclear reactions and by stopping electrons) Radiation belt Cosmic rays, inner rodiotion belts, solar cosmic rays Vicinity of planets and sun (produced in nuclear inter* actions — decoys into pro- ton and electron) Cosmic roys (nucleus of helium atom) Cosmic rays (nuclei of heav- ier otomsl From H. E. Newell and J. E Naugle, Science, 132i 1*65 (1960). Fig. 2.3 Space radiation: cross section of the radiation pattern in longitude 75° west, from Explorer satellites. (Adapted from the New York Times, October 27, 1958.) '9- 2,4 Pioneer V, o 95-1 b highly instrumented space planetoid, was launched March 11, 1960, to supply the first comprehensive data collected in interplane- ,ar y space. 15 o t— < 3 o z o I— u u% in 1/1 § z I s z ■■■'. : '" v -. '■ ' /.-■■■ ■ ■ ■ >. '■-,. i ■■■ ■ ■■■■*■■-*•/■■ ■ •a c y, 5 S-5! £ 5 -n. Dl £ 1 * t I § 1) * a «i - . J! i & Environment of Space 17 in excess of tolerable dosages (Table 2.3), it may be necessary to plan flights from the earth along trajectories which avoid the regions of concentrated radiation (Fig, 2,">). Table 2.3 Maximum permissible radiation dosages and some typical exposure levels {in roentgens) Item Amount Permissib/e exposures Maximum permissible dosages Maximum permissible emergency exposure Typical exposures Normal radiation level (sea level) Undisturbed interplanetary space (cosmic rays) Heart of inner belt (protons) Heart of outer belt (soft x rays) So!ar proton event (protons) Total exposure 0.3* r/'quarter 5.0 r/yr 25 r 0.001 r day 5-12 r/yr 24 r/hr ~200 r/hr 10-10" r/hr 2-400 r E Ei * Limit prescribed for radiation workers. Under this limit the yearly maximum would be 1.2 r. From H. E. Newell and J. E. Naugle, Science, 132: 1465 (1960). 2.7 MORE DISTANT SPACE The sun's nearest star neighbor is Alpha Centauri, which is more than 4 light-years away. (In contrast, the outermost planet of the solar system, Pluto, is only 0.000 light-year from the sun.) The relative brightness of several stars and their distances from the sun are shown in Table 2.4. Insofar as man knows, the universe is infinite. Scattered throughout this void is an apparently endless number of galaxies, each of which contains millions of stars. Some galaxies are them- selves grouped in clusters. The constellation Corona Borealis is made up of some 400 galaxies. In the observable region around us there are an estimated I billion galaxies, with an average dis- tance between galaxies of about 2 million liglil -years. Galaxies usually have the shape of disks thousands of light- years in diameter. The larger galaxies have spiral arms suggesting a pattern of rotation. Our own galaxy, the Milky Way, appears tn have this form. Our solar system is believed to be situated in 16 18 Looking Out: Rockets, Satellites, Space Travel Table 2.4 Some star distances Star Brightness, relative to sun Distance from sun, light-yeors Alpha Centauri Barnard's Star Wolf 539 Sirius A,B Proeyon A Altair Argo Deneb Betelgeuse 1 1 60,000 23 6 8 5,200 6,600 13,000 4.3 6.0 7.7 8.7 11.3 16.5 180 640 650 one of the spiral arm*, about 30,000 light-years from the center. The solar system is moving at a speed of about 1-.0 mi see, but it takes 200 million vears to complete one circuit of the, galaxy. Ylthough presently envisioned techniques may lead to manned exploration in the solar system, they will not suffice for exploring (he vast distances beyond. If as has been estimated, less than 12 per cent of all stars have planetary systems, then nut of some 200 billion stars m our galaxy, there are some billion with planetary systems. One is led to speculate that out of this number there are probably some systems with earthlike planets that may support life. Communi- cation with distant planets of the galaxy is a matter «,t specula- tion only. And bevond our galaxy arc other galaxies at least out to the limits accessible to present telescopes: some '2 billion light- years in all directions. A FILM INTRODUCTION TO SPACE Ut*mB.28wto (I960), National Fiim^r,Ur Ca^la. For vent from Contemporary Films. Inc., 287 West 25 St, New York I, N.Y. Teachers' guide available, SUGGESTIONS FOR FURTHER READING Baker, Robert H.: "Astronomy," 7th e<L, I). Van Nostrum! Company, Inc., Princeton, NJ. t 1989. „ „., Bauer Carl \ : "The Universe bom fee Known to the Unknown, n» Pennsylvania Stale University. Continuing Kdueation, Umvcrsitj Environment of Space 19 Park, Pa., 1982, 54 pp. A manual for adult discussion study groups to which a Guide for the Discussion Leader is keyed, Duncan, John C: ■■Astronomy," 5th ed., Harper it' Row, Publishers, Inc.. Now York, 1055. Hoyie, Fred: "Frontiers of Astronomy." New American Library of Win-id Literature, Inc., New York. 1955. James. .1. N.: The Voyage of Mariner II, Scientific American, 209: 70-84 (1983). Krogdfthl, Wasley S.: "The Astronoinical [.'inverse," The Macmillan Company, New York, 1952. McLaughlin, Dean II.: "Introduction to Astronomy," Houghton Mifflin Company, Boston, 1981. Newell, II. !■",.. and J. K, Xaugle: Uadiulion Environment in Space. Sriri„r. 1:12:1 if 15 I 172 (Mil ill). Shaw, John II.: The Radiation Environment of Interplanetarv Space, Applied Optics, I -. 87-95 (1902). Zim. Herbert, and Robert 1L linker: "Stars." Colder, Press. Xew York 1950. QUESTIONS FOR DISCUSSION 1. Would you expect Mercury to have an atmosphere, that is, a permanent gaseous envelope? 2. Estimate the fraction of the total mass of the solar system which » i" I lie sun. Am. (iO.fi per cent 3. Does a physical environment of the sort needed to support plant and animal life such as we know exist elsewhere in the universe? Where? Can you conceive of a form of life not based on water chemistry? Might ammonia or fluorine compounds serve? Where in the universe would you expect lids differenl form of life to exist? If it docs exist in intelligent. r,, rmi ,[,, vnu tM j n k Wl , ( . ou j ( | ,. tHnmu „i t . at p ,vilh it? I. A point on the earth's equnlor is carried ahoul 1,090 uii.hr by the rotation or the earth. Jupiter has an equatorial diameter II times thai of the earth and a day of 10 hr. Calculate (he speed of a point on the equator of Jupiter. .W _>2.i mi hr 5. Express the diameter of Pluto's orbil in mi. Ami. 7,309 million mi 6. In what ways is it true that all our sources of energy— plant life, "oal. oil, and water -arc derived from the radiant energy we receive uom the sun? T. Furnish some evidence for or against the statements: The climate "ii Mars is similar to that which one would encounter on a iO-mi-high ' ' -'i on earth. Granted transportation, a self-sustaining colon v might '"' established on Mars. 8. The four outermost of Jupiter's 12 satellites revolve about • npitei- rrom east to west, contrary to Ihe motion of most satellites in i solar system and to the direction of revolution of Ihe planets around 1 «m. Can yon suggest a possible reason for this retrograde motion? Rocket Propulsion 21 Rocket Propulsion Necessity is not the mother of inven- tion; knowledge and experiment are its parents. W. R. Whitney All vehicles move !>y reaction with some other thing. Cars require traction on the road. Snip* and planes push or pull themselves through water or air. Only rockets carry their own "something else" to push against. In the words of J. X. Savage, "a rocket is any machine that propels itself by ejecting material brought along for the purpose." A rocket is an internal-combustion engine that carries its own supply of oxygen (in any of several forms of "oxidizer")- There- fore, it does not require air but can operate in a vacuum, as in space. The description of a rocket in flight is a particular application of the general theory of the dynamics of rigid bodies. It is con- venient to consider separately the motion of the center of mass and the motion of the body around its center of mass. The former is the theory of flight performance, the latter, the theory of sta- bility and control. The powered flight of a ballistic rocket is usu- 20 ally, for practical reasons, confined to two dimensions. So the theory of motion in one plane is adequate. In further simplifi- cation, we may begin by considering the flight path to be a straight line (one-dimensional theory). We shall consider in this chapter the basic principles of rocket propulsion; the effects of mass ratio, specific impulse, and 1 1 mist/ weight ratio on the flight of a single-stage rocket; and then the performance of multistage rockets. 3,1 MECHANICAL PRINCIPLES: NEWTON'S LAWS Three laws formulated by Sir Isaac Newton in the seventeenth century are fundamental to rocketry: 1. A body at rest remains at rest and a body in motion continues to move at constant speed in a straight line unless acted upon by an externa}, unbalanced force. 2. An unbalanced force acting on a body produces an acceleration in the direction of the net force, an acceleration that is directly proportional to the force and inversely proportional to the mass of the body. 3. For every force that acts on one body, there is a force equal in magnitude but opposite in direction that reacts upon a second hody. Table 3.1 Consistent systems of units for Newton's second law Name of system Unit of mass Unit of force Unit of acceleration Mks (absolute) kilogram newton* meter, second - Cgs absolute gram dyne* centimeter second - Cgs gravitational No name assigned m = W, g gram centimeter, second 3 British absolute pound poundal* foot/second 2 British gravitational slug* pound foot /second 1 Any system W, g Same unit as that Same unit as that used for W used for g In each set the starred unit is the one usually defined from the second law so as to make k = 1 in F = kma. 22 Looking Out : Rockets, Satellites, Space Travel 3.2 UNIFORMLY ACCELERATED MOTION It is convenient to list and remember the equations which apply to a body which moves with constant acceleration in a straight line. This is a special case, but one often met. The average speed v is the distance traveled divided by the time required, 5 = s/t, or , = U M> Since we have assumed motion in a constant direction, the accel- eration is the change in speed divided by the time, a - (»i — Vi)/t, or v, - „ = at 03) Since the speed changes at n uniform rate, the average speed f is equal to half the sum of the initial and final speeds: »! + "« (3.3) 2 By combining these, two other useful equations can be obtained. Eliminating u s and v, we get s - M + Jo* 1 < 3 - 4 > By eliminating 6 and t from Eq. (3,1) to (8.3), we get S = 3.3 REACTION PRINCIPLE A rocket engine develops thrust by employing Xewton's third law in the following manner. Imagine a stationary sphere (Fig. 3.1«) containing a combustible mixture of gasoline vapor and air. p F (a) (6) M Fig. 3.1 Reaction, the principle of jet propulsion. a Ah- P, Rocket Propulsion 23 If this mixture were ignited, there would result a high pressure p t in the chamber exerting force equally in all directions. The sphere would remain at rest because there would be no net force acting on it. Consider a section to be removed from one side of the sphere so that the gases could escape. The sphere would now experience a net force. Since there would be no balancing force across area A\ (Fig. 3.1b), the force on area A-> would cause the sphere to move to the left. The magnitude of this force or thrust F would be equal to the product of the pressure p c in the chamber and the area A t of the throat: F = pAt cm A greater force can be obtained under certain conditions (Sec. 3.(>) by adding an expansion nozzle at the exit (Fig. 3.1c). The contribution of the nozzle is represented by a thrust coefficient Cp used as a multiplier in the previous thrust equation, so F = PrAtCp (3.7) From Xewton's laws, if F is the net external force applied to a system, the rate of change of momentum of the system is A(mi>) _ „ (3.8) When a rocket is in free space, the net external force acting upon the rocket is zero. If mass particles are ejected from the rocket with a constant exhaust velocity iv, their rate of change of momen- tum gives the rocket an accelerating force ' - " A, * (3.9) The negative sign expresses the fact that F and c, are in opposite directions. The exit pressure p„ of the gas from a rocket often may be either greater or lower than pressure of the racket's environment, ambient pressure p«. Also, while p r remains constant, p„ will de- crease as the missile gains altitude. If the difference between the two pressures is multiplied by the exit area .4,.. we have the mag- nitude of the unbalanced force (p c — p a )A e acting on the rocket. 24 Looking Out: Rockets, Satellites, Space Travel This force is called the "pressure thrust," in contrast with the "momentum thrust" expressed in Eq. (8.9). The total thrust of a rocket engine can be expressed as the sum of the momentum thrust and the pressure thrust: F = - -^T- V* + (}Je - V-) A - (3.10) "Usually, the only term on the right-hand side of this equation that will vary with respect to time is p„, the ambient pressure. 66 64 /-" . 62 S o f X X =e 60 / 1 / £ 58 - / 56 54 i il _! 20,000 40,000 60,000 Altitude, ft 80,000 100,000 Fig. 3.2 Decrease of thrust with altitude, V-2 missile. Positive thrust Negative thrust -t Fig. 3.3 Pressure thrust in a rocket motor. The resulting decrease in thrust as a missile gains altitude is shown in Fig. 3.2. If, as often at sea level, p e < P-, the pressure thrust term will be negative (Fig. 3.3). Rocket Propulsion 25 3.4 EFFECTIVE EXHAUST VELOCITY In order to simplify the thrust equation, an effective exhaust velocity >\. fi is defined as F V,n Am A! Then Eq. (3.10) is written in simplified form as (3.11) _ Am (3.11a) Of course v M is variable with altitude, whereas », is constant for a particular rocket system. Cnder optimum conditions for expan- sion of the gas, when p, = p„, the effective exhaust velocity e,. f r if equal to the theoretical exhaust velocity r,. 3.5 SPECIFIC IMPULSE The performance of a rocket engine is conveniently described by its specific impulse. This is the thrust produced divided by the weight of propellant consumed per second Aw/At (3.12) Since F and w are expressible in the same unit (e.g., pounds), the unit for specific impulse is the second. If other factors are held constant, the speed that a missile can attain is directly propor- tional to the specific impulse of its propellants. The specific impulse varies with altitude, since thrust is vari- able with altitude. By combining Kqs. (3.11) and (3.12), the following useful relationship is obtained: j _ /■' _ V v it Am / At r,. fi Air/ At " i) Am II ' ~f (3.13) To avoid the difficulty of having /, become infinitely large as the gravitational acceleration g approaches zero at high altitudes, it is generally agreed that the value g u = 32.2 ft/sec- si mil be used in K<). (3.13): i t — — (3-14) 26 Looking Out: Rockets, Satellites, Space Travel The simplified graph (Kig. 8,4) b intended to summarize the facts that certain quantities, such as theoretical exhaust velocity u„ propcllant flow rate Am /At, gas pressure in chamber p„ and exit pressure p, are constant for the rocket system. Other quanti- ties, such as thrust F, effective exhaust velocity v Mt and specific F> "•«' 4 -*£.v e ,P e ,P c Sea level Altitude 150,000 ft Fig. 3.4 Simplified representation of the fact that dm, dt, v„ p„ and p t are independent of a rocket altitude, while F, v,ti, and (, increase with altitude since they depend on ambient pressure p a . impulse /„, vary with altitude since they depend upon the ambient pressure p a . Table 3.2 Performance of typical liquid propellant combinations (calculated for expansion from 300 lb in. 1 to 1 otm) Propellant combination Mixture Exhaust Specific ratio velocity, impulse, (oxidizer/fuel) ft/sec sec Liquid oxygen and 75% ethyl alcohol, 25% H 5 (V-2 propellant) 1.3 7700 239 Liquid oxygen and liquid H-,. 5.33 10,800 335 Liquid oxygen and kerosene 2,2 7,970 248 Fluorine and hydrazine 1.9 9,610 299 H-.0;:(S7%) and H,.0 (13%) 4,060 126 Red fuming nitric acid and aniline 3.0 7,090 221 Ni from ethane 7,010 218 From G, P. Sutton, "Rocket Propulsion Elements," John Wiley &. Sons, Inc., New York, 1949. Rocket Propulsion 27 As the measure of over-all engine performance, specific impulse is related to both combustion performance and expansion per- formance. From thermodynamics il may be shown that <-£ (3.15) where T is the combustion temperature and M is the molecular mass of the exhaust gas. Thus a hot, lightweight gas gives a high specific impulse. Roth a large value for the heat of propellant combustion and low specific heat of the gas are desirable to pro- duce the high temperature. The requirement of low-molecular mass suggests that the products of combustion should be rich in hydrogen compounds. 3.6 FUEL COMBUSTION AND EXPANSION The basic principles we have been discussing and some refine- ments in design can be illustrated by considering a typical rocket engine using a liquid fuel and oxidizer (Kig. 3.5). The engine con- Combustion De Laval chamber nozzle Fig, 3.5 Simplified liquid racket motor. verts the thermochemical potential energy of the propellants into the kinetic energy of the gas in the exhaust jet. The steps involved ;nc propellant feed, injection, ignition, combustion, and expan- sion. Tin: liquid propellants arc forced from their tanks into the injector by means of compressed gas or a turbopump. The injector distributes the fuel and oxidizer in a flow pattern that causes thorough mixing. Ignition is started by a device at the surface of the injector; thereafter heat from the combustion gases main- 28 Looking Out: Rockets, Satellites, Space Travel tains continuous ignition. Combustion takes place throughout the combustion chamber with some residual burning in the exhaust gas jet. During combustion, the propellants change from liquid to gas, and by electron sharing they combine to make new chemical compounds. Chemical potential energy is converted into thermal energy, raising the gas to a very high temperature. The change from the liquid to the gas state plus the high temperature of the gas results in a high chamber pressure. Gas particles are forced to the rear. It is the purpose of the nozzle to allow the gases to leave the rocket in smooth flow and also to accelerate these gases. The rear of the combustion chamber first converges to a throat area A t and then expands to an exit area A m which may have about the same diameter as the combustion chamber. The change from potential energy (nondireeted thermal motion of gas atoms) to the kinetic energy of a high-velocity gas jet occurs in two steps. As the gas passes through the converging portion of the nozzle, the decreasing cross-sectional area causes the flow to speed up. The gas flow reaches a maximum speed corresponding to sonic speed (Mach 1)* at the nozzle throat provided the chamber pressure exceeds a critical value, approximately twice the sur- rounding atmospheric pressure. The addition of a diverging nozzle provides for even more acceleration of the gases. A typical throat speed may be 4,000 ft/sec and exit speed 7,000 ft/sec. The expan- sion area rat in A, A t Cilo) is chosen for a particular engine to give the highest average thrust over the powered portion of the trajectory, For a given c a bell nozzle may be some 30 per cent shorter than a conical nozzle, and hence its use conserves rocket weight. An interesting phenomenon called jet separation may add additional thrust. When the exit pressure is very low in compari- son with the ambient pressure, gas flow breaks away from the * Mach number M is defined as the ratio of Free stream speed v to the local speed of sound a, M = v/a\ it is the ratio of directed molecular motion to random molecular motion. Rocket Propulsion 29 wall before reaching the nozzle exit. The thrust coedieient is slightly higher during separation than for a full-flowing nozzle. Jet separation . Optimum expansion Jet separation Under- expansion Pe > Pa ' expansion Pe <Pa Expansion area ratio, 6 Fig. 3.6 Jet separation. 3.7 BURNOUT VELOCITY AND RANGE Consider the case of a rocket moving in a straight line inclined at an angle 8 with respect to the direction of gravity, with thrust F paraDel to the path. The equation of motion will be to m At — F — D + mg cos (3.17) where D is the aerodynamic drag and g is the acceleration of gravity at the location of the rocket. Since D usually depends on Fig. 3,7 Forces on a rocket, the shape and speed of the rocket and the density of the snrround- n»g air, let us assume for this illustration that the rocket is at such 30 Looking Out: Rockets, Satellites, Space Travel high altitude that D = 0. If we divide Eq. (3.17) by m and use Eq. (3.13) to set F = gJ.Am/At, we have 1 Am t'^'irs- (3.18) If we assume tliat the rocket starts from rest, v = 0. We set /« = tVi/ff and let R he the ratio of initial mass to final burnout mass, R = m u /m b . Then E(|. (3.18) can be solved (Appendix ») to find the velocity a at burnout vt = hu In R — gh cos (3.19) where k is the duration of burning. The two averages iv« and g are necessary since the values of both effective exhaust velocity and gravitational acceleration are dependent on altitude. Greater range and less time for interception of a rocket will result from increasing the burnout velocity of the missile. This improvement, can he obtained, according to Eq. (3.19), by increasing the effective exhaust velocity and the mass ratio. 3.8 MASS RATIO The mass ratio is defined as the quotient of the initial or total mass m of a rocket and its burnout mass »h 4 : III:, (3.20) This is one or several dimension less ratios useful in comparing rocket designs. Others, whose- definitions should lie apparent, arc the thrust, full-weight ratio, the dead-weight fraction, and the payload fraction. The burnout mass is related to the initial mass simply by (3.21) Am , where Am /At is the propellant flow rate, From Eqs. (3.18) and (3.19) it is evident that to achieve the desirable high burnout velocity, a fuel with high specific impulse is needed. Further, for a given value of /„ larger mass ratios pro- vide higher values of iv The mass ratio of the World War II German V-2 rockets was about 3.2. For present rockets R U as high as .">. Rocket Propulsion 31 3.9 MULTIPLE-STAGE ROCKETS In a single-stage rocket the propulsion energy must be used to accelerate the entire empty mass of the rocket even after most of that empty mass is no longer useful. This severely limits the velocity attainable. Tn fact, with present fuels, a single-stage rocket cannot achieve velocities of the order of 25,000 ft/sec and higher required to place a satellite in orbit or to escape the earth's gravitational field. A multiple-stage rocket is made up of a number of independ- ent sections each equipped with a propulsion system and a portion of the total propellant load. After the first (booster) stage has lifted the entire rocket and has reached its burnout velocity, its empty mass is dropped from the rocket. A second (sustainer) stage carrying the payload is then fired and continues to accel- erate the now lightened missile to the appropriate final velocity. ( )f course more than two stages can be used, but design and oper- ational difficulties become more numerous as stages are added. If each of a series of stages has the same values of specific impulse, dead-weight fraction, payload fraction, and thrust/ weight frac- tion, each will contribute the same increase in velocity to the final payload. This design results in the lightest over-all rocket to perform a given mission. A simplified expression for the burnout velocity of a two-stage rocket is Vt = fvrln (/r,/rs) (3.22) Here R t is the initial mass of entire rocket divided by the burnout mass of the first stage plus the initial mass of the second stage, R = (wioi + v»us)/("'m + »t 02) and R.< m hi,,., ,jj,,,. if the second stage is made small in relation to the first stage, the value of the logarithmic term in Eq. (3.22) will he greater than that in Eq. (3.19), predicting a greater final burnout velocity for the two- stage rocket than that given by Eq. (3.19) for a single-stage rocket in vertical flight, namely, 1% = i\.rr In R. 3-10 NUCLEAR PROPULSION Some advanced concepts for rocket-propulsion systems have to do with development of recoverable boosters, restartable engines, s1 "nit>k' propcilants. and nozzles which allow a reduction in 32 Looking Out: Rockets. Satellites, Space Travel engine size. But efforts are also being made to find new sources of rocket power, other than chemical reactions;. Figuratively, we should like to be able to pack the power output of Hoover Dam (1.3 X 10 6 kilowatts) into a sports car. The development of nuclear power sources promises to provide specific impulses sig- nificantly greater than the values, around 400 sec, for chemical fuels. Research on the use of a nuclear reactor as a rocket energy source has been carried out since 1955 in Project Rover, directed by the Atomic Energy Commission and the National Aeronautics Pressure shell Nozzle Reactor core Fig. 3.8 Scheme for a nuclear-powered rocket engine. and Space Administration. The test engines have been named Kiwi's, after a flightless bird. Heat is generated in solid-fuel elements by nuclear fission (('hap. 20). Hydrogen gas flows through channels in the core. The heated gas is exhausted at high speed through a nozzle (Fig- 3.8). The thermodynamic (Caniot) efficiency of any heat engine is given by 7', ■ T-> Ti where 7'i is the temperature (absolute) of the source of energy and Ta is the temperature at which the working fluid is dis- charged. The lieat -exchanger nuclear engine exhausts into a relatively low temperature environment, especially when in Efficiency = (3.23) Rocket Propulsion 33 Fig. 3.9 KiwI-A nuclear engine at Project Rover test site in Nevada. space. So one would expect to be able to put almost all the nuclear energy into thrust. The limiting factor is the energy den- sity one can put into the propellant to eject it at sufficiently high speed. \ssumc thai one has an ideal nozzle to recover directed kinetic energy from the thermal motion of the propellant mole- cules and that the propellant acts as a perfect gas. Then W RT (3.24) and the exhaust velocity v c is proportional to \/T/p or to \/T/M, where p is the density of the propellant, M is its molecular mass, and Ft is the universal gas constant. For high velocities one wants maximum temperature and minimum molecular mass. Thus hydrogen heated to the highest feasible temperature gives the largest specific impulse of any material. Estimates range from ,J 00 to 1,500 sec for the specific impulse of a heat -ex changer 34 Looking Out: Rockets, Satellites, Space Travel nuclear rocket engine. Recalling the relation n, = 7» p ln (mo/m,), it is apparent that the larger l. p attainable with nuclear pro- pulsion allows one to reach a desired orbital velocity or escape velocity with a much lower initial fully fueled weight (smaller mass ratio )«<>/'«»)• Under Project Sherwood, studies are being conducted to find ways of controlling and using the energy liberated in the fusing of the lightest nuclei into heavier nuclei. The phenomena involved in thermonuclear (fusion) rockets, plasma rockets, and ion rock- ets fall under the general term magn^ohydrodynamica: the study of the behavior of ionized gases acted upon by electric and mag- netic fields. Deuterium is a likely fuel for a fusion rocket. Heated to a very high temperature, the deuterium would maintain a high-speed plasma (hot, ionized gas) capable of specific impulses rated in millions of seconds. There is a difficult problem in confining a plasma at the temperatures estimated to he around :i">0 million degrees. Perhaps the plasma could be kept from coming into con- tact with material walls in chamber and nozzle by suitably shaped magnetic fields. 3.11 ION PROPULSION The removal of one or more electrons from molecules of a propel- laut, by passing the propelknt through heated metal grids, pro- vides ions which can then be accelerated to high velocities through a nozzle by an electric field. Volt age takes the place of tempera- ture in producing acceleration. One such technique uses metallic rubidium or cesium prope.l- lant and tungsten grids. Each time an atom of cesium comes in contact with the heated tungsten grids, an electron leaves the cesium atom and goes to the tungsten metal. The resulting cesium ions travel past decreasing potential levels and arc accelerated to their final exhaust velocity. The ion rocket will always have relatively small thrust. It will require assistance (from chemical or nuclear rockets) in ground takeolTs where strong gravitational force must be over- come. But the performance of an ion engine at high altitudes will be very good. Estimates of its specific impulse are as large as 20,000 sec. The amount of electric power required for an ion Rocket Propulsion 35 rocket is very large. The weight of the electric power plant, even using nuclear fission or solar radiation devices, is a major obstacle to an efficient ion rocket. Electrons Distributor plots and housing ot 40,000 volts >. s El eel V-a— i Electric generator f Electrons Propel I ont Heater coils ^Ionization gr 20,000 volts-'' 10,000 volts- 5,000 volts Fig. 3.10 Scheme for an ion rocket engine. Propellent injection Arc discharge High- current circuit Fig. 3.11 Scheme for plasma rocket engine. 3.12 PLASMA PROPULSION Hie propcllant may be heated directly by maintaining a powerful electric arc in it. In this way high temperatures can be obtained, leading to a specific impulse of perhaps 2,000 sec. But this device, 36 Looking Out : Rockets, Satellites, Space Travel too, will require 8 great amount of electric power, about (50 kilo- watts for each pound of thrust. In plasma and ton propulsion the thrust can be applied con- tinuously over an extended period of time. Hence by these tech- niques one can propel in space rockets whose weight on earth greatly exceeds the thrust of ion propulsion. 3.13 SOLAR PROPULSION In one scheme of solar propulsion, the radiation pressure of solar rays (ailing on a "sail," perhaps a lightweight reflecting sphere, attached to the spaceship would propel if. In another scheme, the Sun's rays heat water circulating at Focus of mirror, producing steam Steam-driven turbogenerator to develop Solar sail Fig. 3.12 Schemes for solar propulsion, (o) Steam generated by solar energy drives electric generator. {b\ Recoiling photons impart momentum to sail. Rocket Propulsion 37 solar rays would he used to heat hydrogen gas which would then lie expelled through a nozzle. Kstimated values of the thrust are low hut are several hundred times those of an ion or a fusion system. 3.14 MODEL ROCKETRY Many a youth has felt the urge to become a backyard rocketeer. The National Association of Rocketry, founded in 15)57, seeks to advance model rocketry as a scientific hobby and as ati edu- cational program. The NAR has developed rules and procedures for a safe, supervised, citizen-operated model-rocketry program for enthusiasts of all ages. Model rocketry is concerned with small, light, inexpensive rockets made of paper, balsa, plastic, and other noninetallic materials, powered by commercially available rocket motors. Emphasis is placed upon design, performance, flight character- istics, instrumentation, and reliability. Competitions are spon- sored hy local societies. I'iiblicat ions ami informal ion about the XAR can lie obtained from G. Harry Stine, President, National Association of Hock- el rv, Stamford Museum it Nature ( "enter, Stamford. Connecticut. Physics teachers may he interested in model rocketry as a device for stimulating student interest in mathematics, mechan- ics, aerodynamics, meteorology, electronics, optics, and pho- tography. Ideas based on the experience of the most active sections of the NAIl may be requested from Dr. Stine at the address above. MANUFACTURERS OF MODEL ROCKETS American Telasco Limited, 135 New York Ave., IhUesitc, X.V. Centuri Knpncering Co.. 340 \V. Wilshirc Drive, Phoenix 3, Ariz. Ci>a»ler Corporation, P.O. Box 2S0, Hiiless. Tex. Bates Industries, Inc.. P.O. Box 227. Penrose, Colo. Model .Missiles, Inc., 2ti!<) Bast Cedar Ave., Denver 22, Colo. Propulsion Dynamics. Inc., P.O. Pox 2XXA. Ut. 1, Officii, Utah Rocket Development Corp., Box 522, Rich mood, bid. Cnited Scientific Co., Inc.. P.O. Box S9, Waupaca, Wis. FILM The Itislor,/ ami Development of the fiwkri. 10 min (1962). MeUruw-llill Text-Film Division, 330 West 42 St., New York 36, N.Y. Available in color or black and white. 38 Looking Out: Rockets, Satellites, Space Travel SUGGESTIONS FOR FURTHER READING Hobbs. Marvin: "Fundamentals of Rockets, Missiles, and Spacecraft," John F. Rider, Publisher, Inc. New York, L962, 27ft pp. "An Introduction to Rocket Missile Propulsion," Rocketdync, Canoga Park, Calif.. 1958, 12ft pp, "Model Kits," Revcll, Inc., 4223 Cileneoe, Venice, Calif. "The Next Ten Years in Space, 1959-1989," House Document 115, U.S. Government Printing Office, Washington 2ft, DC., 1959, 221 pp. , "1959 Missiles and Rockets Encyclopedia," Re veil, Inc., Venice, Calif., 32 pp. "Physical Data, Constants and Conversion Factors," General Electric Missile and Space Vehicle Department (Mail: P.O. Box Sftfto, Philadelphia 1, Pa.), 1959, 24 pp. •■Rocket Experiment Safety: Safety Suggestions for the Rocket Hobby- ist/' Atlantic Research Corp., Alexandria, Va., 1958, 19 pp. Seiferl, Howard S.. Mark M. Mills, and Martin Summerlield: The Physics of Rockets, American Journal of Physics, 15:1-21, 121-140, 255-272 (1947). "Space Facts: A Handbook of Basic and Advanced Space Might and Environmental Data for Scientists and Engineers," General Elec- tric Mis>ile and Space Vehicle Department, Valley Forge Space Technology Center (Mail: P.O. Box 8555, Philadelphia 1, Pa.), I9f>0. fil pp. QUESTIONS AND PROBLEMS 1. Verify the statement: "Near the surface of the earth, gravity robs a vertically rising rocket of about 20 mi/hr in speed each second, or about 2,400 mi/hr for each 2 iniu of acceleration." ^ 2. A projectile is Srcd with a speed of 300 ft/sec at an angle ot 87 with the horizontal. Compute the speed when it first reaches a height of .jqq- fj._ j4«s. 83.5 ft, sec 3. A force of 4,900 dynes acts on a 20-gm mass for 8.0 sec. (a) What acceleration is caused? (I>) Bow Tar docs the mass move from rest in the 8.0 sec? («) How fast is it going at the end of S.O see? Am. 24ft em/see, 8,140 cm, 1,900 era/see 4. Show that the mass ratio tin/m,, for a multistage rocket is the product of the mass ratios of its individual stages. 5. Would you consider an alkali metal, such as cesium, a prospec- tive propellant for an ion rocket? Why? 6. If it becomes possible to convert nuclear-fusion energy of a plasma directiv into electric energy, without the conventional rotating gener- ator, would this make ion propulsion of rockets more feasible? Rocket Propulsion 39 7. Show that if one increases the exhaust temperature of a hc-al- cNclianger nuclear rocket, the specific impulse and power requirements will increase as J' ! and the mass ratio will decrease as exp T. What limits this favorable picture? 8. The Atlas 1CBM is called a I '-stage rocket because of its unique application of the step principle. The Atlas has three main engines: two booster engines and one suslainer. Each engine receives propellant from a single very lightweight tank. The three engines are mounted parallel to one another. All three engines are ignited at take- ofT. I/titer. at staging, the boosters ami housing slide backward on rails and drop to earth, leaving the suslainer engine to propel (he vehicle. Can you suggest some advantages of this type of staging over the con- vent iona! tandem staging? 9. How can rocket action be demonstrated with a toy balloon? 10. What is the fallacy in the following argument? "A horse pulls on u cart. By Xewton's third law, the cart pulls back on the horse with a force etjiuil to that exerted by the horse on the earl. Hence the sum of the forces is zero, from which it follows that it is not possible for the horse to accelerate the cart." 11. Comment on the remark, "Space stations will be obsolete when they are feasible." 12. A rocket whose thrust is 27,000 lb weighs initially 22,000 lb, of which SO per cent is fuel. Assuming constant thrust, find the initial acceleration and the acceleration just before burnout. Xeglect air resistance and variation of g. Arts. 39.2 ft/sec 2 , 190 ft/sec 2 13. From Eq. (3.19) show that for a rocket launched horizontally and continuing in a path parallel to the earth, the burnout velocity is given by Vt = err In R. II. What is the minimum value of mass ratio R for which the burnout velocity n, of a rocket will exceed the effective exhaust ve- locity iv„? 4 ns. R > 2.718 15. Do you agree with Professor Fink's comment that Hie methods of achieving lift listed in order of increasing sophistication of the under- lying physical principle are (1) satellite vehicles, (2) displacement of lighter-than-air craft, (3) hover craft or ground-cushion vehicles, (4) vertical flight rockets, (5) vertical takeoff and landing machines, (<>) conventional airplanes? If so, how do you account for the historical fact that the "simplest" methods were not the first to be suecessfullv Used? Escape from Earth No thing is too high for the daring of mortals : We storm heaven itself in our folly. Horace Every great advance in science has issued from a new audacity of imagi- nation. John Dewey Through the ages men have dreamed of the power of flight. In Creek mythology Daedalus and Icarus made a daring ascent into the air on wings made of birds' feathers and wax. In the notebooks of Leonardo da Vinci are found detailed drawings of a flying machine. With the success of the Wright brothers, man began to realize his long ambition of flight through the air. But now he turns his dreams to flight beyond the enveloping and protective atmosphere — into space. In designing a vehicle to escape the earth, one has to solve the problem of piercing the earth's atmosphere. A second, more troublesome, problem of escape is that of overcoming the force of gravity. Since each body in the universe has its own gravita- 40 Escape from Earth 41 tioual field, a vehicle in space would encounter an endless mixture of gravitational fields, one superposed on another. The terms escape and capture refer to the transfer of the vehicle from one field to another. 4.1 GRAVITY In addition to the three laws of motion, Newton formulated the law of universal gravitation: Every particle in the universe attracts every other particle with a force that is directly propor- tional to the product of the masses of the two particles and in- versely proportional to the square of the distance between their centers of mass F = G (4.1) where F = force of attraction m [ and mi = masses of the two particles s = distance between them G = gravitational constant, whose value depends on the system of units used In mks units G = (5.07 X 10 " newton-mVkg s Gravity acts as a brake on a vehicle leaving the earth. While traveling in space, a vehicle is always subject to some gravity. The vehicle attracts and is attracted by all celestial bodies. But because gravitational force follows an inverse square law (/•' <* 1 «-), the mutual attractions of only the nearest bodies are usually significant. When a vehicle returns to earth, it is accelerated by an increasing gravitational force. 4.2 FREELY FALLING BODIES An unsupported body starting from rest near the surface of the earth drops 10 ft during the first second, 04 ft at the end of the iH'Xt second, 144 at the end of the third, etc. It has an acceleration r, f "V2 ft per sec per sec, or 32 ft/sec-. The symbol g is used to represent the acceleration due to gravity. At sea level and 4:">° latitude, g has a value of 32.17 ft/sec 2 or 9.806 m/sec 2 . The value of g varies slightly over the earth owing to local 42 Looking Out: Rockets, Satellites, Space Travel variations in mass distributions and to the fact that the earth bulges slightly ai the equator. Surface gravity values vwy from planet to planet owing to differences in mass, radius, and rota- tional speed. When air resistance can be neglected, the equations in Sec. 3.2 for uniformly accelerated motion apply to falling bodies. t, sec 1 v, ft/sec s, ft O O °\ 32 64 96 128 16 64 144 256 Fig, 4.1 Position and speed of a body falling freely from rest after successive equal time intervals. 4.3 GRAVITATIONAL FIELDS The force which one body exerts on another at a distance is con- veniently described by the "force field" set up by one of the bodies, the source. Various kinds of forces can be treated in this way. Electric charges exert forces upon other electric charges. Magnets exert forces on other magnets. Matter exerts gravita- tional force upon other matter. The force exerted on a unit test particle (unit charge, unit mass, etc.) has a definite magnitude and direction for each pos- sible location of the test particle. The whole assemblage of these for08 vectors, or the mathematical function relating force to position, is called a field of force. Any path that would be fol- lowed by a free incrtialess test particle is called a line of force. The gravitational field intensity / at any point A in the space near a mass ,1/ is defined as the force per unit mass acting on any mass m placed at A : / = m (4.2) Escape from Earth 43 The small mass m is used only as a means of detecting and measur- ing the gravitation field, Whether m is large or small, the. force per unit mass placed at .4 has a definite value, J. By substituting Eq. (1.1) for F in Kq. (4.2), we get the expression for the gravitational field intensity / at a distance r from niitss .1/ in terms of the universal gravitational constant Gas m r 2 (4.3) When one knows the field intensity, one can find the force acting on any mass m as the product of m and /, F Fig, 4.2 Gravitational forces of attraction. Fig. 4.3 Parabolic poth of a projectile in a uniform gravitational field. For a freely falling body, Newton's second law of motion becomes F = mg. At the earth's surface, therefore, the gravita- tional field intensity is equal to g, the acceleration due to gravity ' = - = ? m I newton (4.4) meter kilogram second 2 111 mka units, g = 9.80 m/sec 2 44 Looking Out: Rockets, Satellites, Space Travel In a region of free space where the gravitational field is prac- tically constant in direction and magnitude, the path taken by a projected mass m is a parabola (Kig. 4.3). 4.4 GRAVITATIONAL POTENTIAL ENERGY To find the work needed to get off the earth, let us calculate the work done in moving a mass m from the surface of the earth, radius R, to a distance r from the center of the earth. Imagine Distance r * measured _ from C Fig. AA Calculation of gravitational poten- tial energy. the distance from 5 tor to be divided into small equal intervals so that over each the gravitational force F a will lie practically con- stant. Then we can easily calculate thr work done in each interval and add to get the total. At the surface, Fa = QMjm A' 1 '. At the top of the first interval Fa is OMjm r,«. Since these values are nearly the same, we can use for the average force in the first interval (!M<w ltr u The work done in the first interval is then Wt = Fair, ~ 8) = Vg fir. - *> = W* (l " ,') Likewise the work in the second interval is „,. 9£ h -„-«.„.„, (i-I) Escape from Earth 45 and in the third Wt = GMjn (- - -) If we add these three expressions, the intermediate values r, and n cancel out. The work done in the first three intervals can be expressed in terms of the values of r at the ends: It and r 3 . Thus W = GM - m (it ~ ' ) (4.5) is the general expression for the work required to move a mass in against the earth's gravitational field out to a distance r. By definition this is the gravitational potential energy of mass m in the field of the earth. 4.5 VELOCITY OF ESCAPE, FROM CONSIDERATION OF ENERGY To estimate the maximum height attained by a rocket fired straight up, we may equate its kinetic energy at burnout to the gravitational potential energy it acquires thereafter in rising, with decreasing speed, to its maximum height, t x ^mv 2 = mgR* \lt ''i..:. </ (4.0) Example. A roirkci has an upward speed uf 5.0 mi/see :it burnout. Find the maximum height it attains. From Eq. (4.6) mf-n mi Y m ft (*ixm ■ 3960- mi' \ V 880/ "»* \ r.„n< ) 25 / miV / 32 \ .. i(irin { iniV ( 82 \ /39(i0= mi A %\&) =Uoj c ^ o) tcj -U»/U.*W r n , ax = 8,S(W mi (What amplifying assumptions have been made in this Bohition?) 4 -6 VELOCITY OF ESCAPE, FROM CONSIDERATION OF FORCE If a gun on a cliff overlooking the ocean fires a bullet horizontally, the bullet will strike the water at some distance from the base of ■he cliff. If the initial speed of the bullet is increased, the range 46 Looking Out : Rockets, Satellites, Space Travel is increased. For a particular speed, which depends on the dis- tance of the gun from the center of the earth, the hullet would make a complete circuit of the earth, at a constant altitude, A (Fig. 4,">), If it- did not encounter resistance, it would continue to move in orbit about the earth. Prom Newton's second law, F = ma, and Eq. (4.2), the force needed to hold the bullet in a circular path at altitude A is Fig. 4.5 Range increase* with horizontal firing speed until circular (orbital speed is reached. inv-fQi + R). This force is provided by the gravitational attrac- tion of the earth, so /' - CM ,i» i-ni (/,' + /,)■' H + h (4.7) Example. At what speed would a projectile have to leave a platform, horizontally, 300 mi above the earth in order to enter a state of "con- tinuous fall" around the earth? From Kq. (4.7), C.U e 6.67 X IP"" newton-m* 5.983 X 10" kg ys = - R + h kg' = 58.1 X ICm'/sec* w = 7,«2() in sec = 1,700 mi/hr mi (3,950 + 300) mi 1,609 m 4.7 WEIGHTLESSNESS A body in orbit around the earth or following an unrestricted, un powered course in a gravitational field anywhere in space is said to be in "free fall," also called a state of "zero gravity." Actually, gravity is not absent. The force of gravity continually Escape from Earth 47 acts on the body and determines its path. But the condition of weightlessness is experienced because there is nothing to resist the body's motion in response to gravity. Human beings have experienced weightlessness for the first few seconds after leaving a high diving board, or for somewhat longer periods in aircraft on "zero g" trajectories, and more recently in manned rocket flights. 4.8 POTENTIAL-WELL MODEL Using Eq. (4."»), we may plot a graph showing the potential energy K p which a body of mass ni would have at various dis- tances r from the center of the earth (Kig. 4.6). When the mass m 50,000 25.000 ,,„, ennnn ■ S GrOVltOtlOnol 25,000 50, 000 mi f , 1 1 £■ free •■ : spoc Fig. 4.6 Gravitational potential energy of mass m, showing "well" analogy for earth's field, is infinitely far from earth, li p = 0. As mass m is brought closer and closer to the earth, work is done on m by the earth's field and the potential energy of m acquires a larger and larger negative value. Thus on the surface of the earth we live in a gravitational well thousands of miles deep. To reach the moon or another planet we must climb out of this well onto the plane marked "gravitational free space" in Fig, 4.(i. A potcntial-energy-well model for demonstrating satellite orbits may be made from a suitably shaped wine glass to reprc- 48 Looking Out: Rockets, Satellites, Space Travel sent the surface (Fig. 4.7) obtained by rotating the graph of Fig. 4.6 ahout its vertical axis. A marble representing the satellite may be caused to travel a variety of orbits by varying its initial velocity. Fig. 4,7 Potential -well model for demonstrating circular (c) and elliptic (e) orbits. ({See J. S. Schooley, Satellite Orbit Simulator, American Journal of Physks, 30: 531-532 (1962).] QUESTIONS AND PROBLEMS 1. What is the largest gravitational force of attraction between two solid metal spheres cadi of 50.0 kg mass and 10.0 em radius? How does lllis force compare with the force of attraction of the earth on each sphere? Ans. 4.17 X 10"° newton, weight is 120 million times larger 2. What would be the value of <j. the acceleration clue to gravity, if the earth had half its present diameter? 3. If the mass of the moon were doubled hut the orbit remained the same, what would he the period of the moon? t. A 100-lb man starts sliding down a rope with a downward acceleration of p/S. (a) What is his apparent weight? (b) What is the tension in the rope above the man? 5. Using the experimentally determined value of (7 and the distance 93 X 10° mi from carlh to sun. calculate the mas* of the sun. 6. At what point in its trajectory does a projectile have its mini- mum speed? 7. If a rocket at tains a speed of (500 mi hr hy the time it reaches 1,000 ft, how many times g is its acceleration? 8. The earth revolves about the sun in a nearlv circular orbit Escape from Earth 49 (r = 150 X 10 e km) with a speed of about 30 km see. What is the acceleration of the earth toward the sun? 9. Show that to escape from the atmosphere of a planet, a molecule of gas must have a speed r such that t ! > 2C.i//r, where .1/ is the mass of the planet and r is the distance of the molecule from the center of the planet. What hearing does this have on the composition of (lie atmos- phere surrounding the earth and other planets? 10. A balloon which is ascending at the rate 12 m/sec is 80 m above the ground when a -lour is dropped from it. How long a time will be required for the stone to reach the ground? Ans. 5.4 sec 11. An elevator is ascending with an acceleration of 4.0 It sec 2 . At the instant its upward speed is K.O Ft/see, a holt drops from the top of the cage 9.0 ft from its floor. Find the time until the holt strikes the floor and the distance it has fallen. Ans. 0.71 sec, 2.3 ft 12. A body hangs from a spring balance supported from the roof of an elevator, (a) If the elevator has an upward acceleration of 4.0 ft/ sir- and the balance reads 45 |b, what is the true weight of the body? (ft) In what circumstances will the balance read 35 lb? {<■} What will the balance read if the elevator cable breaks? Ans. 40 lb, a = 4.0 ft/sec s downward, zero 13. If the mass of the moon is ^ the mass of the earth and its diameter is -J- that of the earth, what is the acceleration due to gravity on the moon? How far will a 2.0 gm mass fall in 1.0 .-cc on the moon? Ans. I g, 3.2 ft 14. A girl standing on a diving hoard throws a ball with a hori- zontal velocity of 50 ft/sec to a man in the water. In doing so, she loses her balance, falls off the hoard, and strikes the water in 2.0 sec, (a) How far is the man from the base of the diving board? (b) How high is the diving hoard above the water? (e) What is the velocity of the ball at the end of its path? .bis. 100 ft, 64 ft, SI ft/scc, at 52° with the horizontal Satellites 51 Satellites It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications, A. N. Whitehead The launching of the first artificial earth satellites, the Russian Sputniks I and II, in 1957, aroused worldwide interest in the power and control attainable with rockets. The special scientific investigations made during the International Geophysical Year were significantly aided by data from instruments carried in satellites. From the orbit of a satellite one may better estimate the shape and dimensions of the earth. A permanent satellite can be useful as an aid in the navigation of ships, aircraft, and missiles. For a satellite which eventually returns to earth, measurements of the orbit may yield a more precise value of g. Atmospheric drag and the effectiveness of radio emission at various altitudes can be studied. Equipped with suitable instruments, a satellite can also measure solar and cosmic radiation, temperature and pressure variations, and the distribution of the earth's magnetic field. In short, satellites can tell us much that we want to know about our earth and much that we need know about space hazards before we venture into space ourselves. 50 5.1 ELEMENTS OF AN ORBIT To define the position of an earth satellite in the solar system and to describe its path, one needs to know the period of the satellite and the elements of its orbit, that is, the constants which fix its position and shape in space: The period is the time for a satellite to make one revolution around the earth. The perigee is the position of closest approach to the center of the earth. Apogee is the position of the satellite farthest from the earth (Fig. 5.1a), The eccentricity describes the flatness of the orbit as the ratio of e to a (Fig. 5.1/j). Here e is the distance from the center of the Apogee Perigee Fig. 5.1 Elliptical orbit. orbit to the focus at the center of the earth, while a is the semi- major axis. The angle of inclination i of the orbit is the angle between the plane of the orbit and the plane through the equator (Fig. 5.2). The plane of the satellite orbit intersects the equator plane in a straight line called the line of nodes. This line intersects the satellite orbit at two points, called nodes. At one of these, the ascending node, the satellite crosses northward from "below" the equator plane to "above" the equator plane. At the other, the descending node, the satellite crosses southward from "above" th f equator plane to "below" the equator plane. The orbit ele- ment we now define is the longitude V, of the node, or the angle of "■•<>•' tiding node. This angle, P. in Fig. ,5.2, is measured in the plane J f the equator from the direction of the vernal equinox to the unction of the ascending node. (To describe the motion of a -Q o Satellites 53 planet about the sun, one substitutes "ecliptic" for "equator" in the definitions above.) The argument of perigee us is the angle measured in the orbit plane between the direction of the ascending node and the direc- tion of the perigee. To summarize, the elements of an orbit are period, perigee apogee, eccentricity, angle of inclination, angle of ascending node, ami argument of perigee. 5.2 CIRCULAR ORBIT The path of any body acted on only by an inverse square force (/-' a l/r ! ) due to a neighboring fixed body will ho an ellipse, circle, parabola, or hyperliola (Chap. (i). To simplify our analysis, for the remainder of this chapter we shall examine an earth satellite in a circular orbit and consider only the interaction be- tween the earth and the satellite. Although small perturbations may be produced by the atmosphere, the moon, other planets, and satellites, for the present these effects will be neglected. For a satellite in circular orbit, the gravitational force exerted on it by the earth has no component in the direction of motion which could either increase or decrease the speed of the satellite. It orbits at constant speed. The force on the satellite is given by Xewton's law of gravitation F = G Mm (5.1) where the mass of the earth M and the mass of the satellite m are icffiirded as concentrated at the center of each, a distance r apart. The constant of gravitation G can In* determined in the labora- tory. Because the mass of the earth is so very large, the center of mass of the two bodies is practically at the center of mass of the earth. The motion may be described as a circular motion of the satellite about a fixed center of force. The direction of the velocity of the satellite in circular orbit » continually changing (Fig. 5.3). Gravitational force continu- ally produces a "centripetal" acceleration a toward the center. 54 Looking Out: Rockets, Satellites, Space Travel Satellites 55 For this uniform circular motion the acceleration is v-/r and the centripetal force is mv*/r. / y — -\A v ¥ canst. \13\- const. \a\— const. Fig, 5.3 A satellite in circular orbit is continually accelerated toward center of orbit. 5.3 PERIOD We may equate the gravitational force and the centripetal force f ,Mm _ mv' 1 G — -z — ~rr (5.2) From this equality we find the speed that a satellite has to obtain to maintain a particular altitude GM v = x - (5.3) The mass m of the satellite does not appear in the equation for speed. The closer the satellite is to the earth, the greater must be the speed, because the gravitational attraction is greater. Since the angular speed tn is 2w/ "period, 2x JOM and since we obtain an equation. 7'2 = GM (5.4) which says that the square of the period of the satellite is propor- tional to the cube of its distance from the center of the earth. This is Kepler's third law of motion, for the special case of a circular orbit. 5.4 ENERGY The total energy remains constant in satellite motion. This can be shown very easily for our special case of a circular orbit. Substitution for the speed of the satellite (Eq. 5.3) into the equa- tion for kinetic energy E k gives v i— * GMm h k = \mv* = - 2r (5.5) Since the gravitational potential energy Ey of the satellite is GMm B, = - 1 1 ie total energy is E k + E„ «■ - GMm 2r (5.6) (5.7) The total energy is negative for both circular and elliptical orbits. This means that the satellite is bound to the center of force and cannot escape unless sufficient positive energy is provided (see *'ig. 4.7). 5.5 ANGULAR MOMENTUM Iho total angular momentum L of a satellite moving at constant speed in a circular orbit is the product of its linear momentum mv 56 Looking Out; Rockets, Satellites, Space Travel and the radius r. Vcctorially, the angular momentum is repre- sented by a vector L drawn to scale to represent the scalar mag- nitude mvr and drawn along a line perpendicular to the plane of r and v in the direction indicated by tlie thumb of the right hand when the fingers are allowed to curt from the direction of r into the direction of v. Thus L results from a vector "cross product," the notation for winch is L = rX (hit) It is obvious that for a satellite in uniform circular motion the total angular momentum is constant. This is also true when a *S^ Fig. 5.4 Angular momentum of mass m is re- presented by vector I. satellite moves in an elliptical orbit. The radius and speed vary, but the total angular momentum remains constant. This is equivalent to Kepler's second law, that a line joining the focus and the satellite sweeps out equal areas in equal periods of time (Fig. 5.5). Fig. 5.5 The satellite sweeps out equal areas in inequal periods of time. It is not feasible or even particularly desirable to launch a satellite into a perfectly circular orbit. If such an orbit were attained, slight perturbations would soon make it elliptical. Observation of a satellite in orbit gives us information about irregularities in the shape of the earth. As the satellite orbits, the plane of its orbit rotates or regresses toward the west. At the same time the orbit turns in its own plane, swinging the perigee around. Also, both ends of the orbit become somewhat flattened. Satellites 57 These observations are interpreted as proof that the earth bulges slightly around the equator, owing to the earth's rotation. The gravitational force tends to pull the satellite toward the equator. Consider the gyroscopic property of the satellite. The gravitational force due to the bulge tends to tip the axis of the Earth's rotation Fig. 5.6 Rotation of the earth and precession of the satellite orbit expose different areas of the earth to the satellite, os shown in Fig. 5,7. orbit. The reaction causes the plane of the orbit to prccesK around the earth in a westerly direct ion, while the earth is rotating from "est to east. This precession may be an advantage in the case of Certain types of observational satellites which thus may "see" most of the earth's surface (Kigs. 5.6 and 5.7). Satellites 59 QUESTIONS I. Why is tin* upper (dotted) path in i)k> accompanying sketch not. a possible satellite orbit about the earth? Fig. 5.8 2. Show ilial if frictional forms cause ;i satellite to lose total energy, it will move into an orbit closer to (he earth with an actual increase in speed. 3. After a certain satellite was put in orbit, it was stated thai the satellite would not return to earth but would burn up on its descent. I low is this possible, since it did not burn up on ascent? Motion of Bodies in Space 61 Motion of Bodies in Space If I have seen farther than ethers, it is hy standing on the shoulders of giants. An old saying quoted by Newton A space vehicle when not under power is governed by the same laws which determine the motions of stars, planets, and comets. These laws are Newton's law of universal gravitation and Kep- ler's taws of planetary motion. Karly in the seventeenth century, Kepler by inductive reasoning formulated his three laws to fit the astronomical observations and calculations made available 1 1! him by his patron Tycho Brahc. Xewton in his "Principia Mathematica" (Ih'87) showed that the kind of planetary motion described by Kepler's laws can be deduced from the universal law of gravitation. 6.1 KEPLER'S LAWS Kepler's description of planetary motion may be stated as follows; Law I. The planets move in ellipses having a common focus situated at the sun. Law II. The line joining the sun and a planet sweeps out equal areas in equal periods of time. Law III. The square of the period of a planet is proportional to the cube of its mean distance from the sun. An ellipse may be constructed by using two pins and a loop of string to guide a pencil (Kig, 0. 1). This method of construction Fig. 6.1 ellipse. Construction of an makes use of a geometrical property of the ellipse: The sum of the distances from any point on an ellipse to the two foci, A and li. is constant. An ellipse with its center at the origin of coordi- nates and with foci on the x axis is represented by an equation <>!' the form + £- = 1 a 2 T b* From Kepler's second law, if the shaded areas in Kig. (>,2 arc S3 x, f~~^ Fig, 6.2 Law of areas. a " equal, a planet takes equal time intervals to travel the dis- tances St, $ 2l and s a . Kepler's third law, called the harmonic law, expresses the Proportionality of period squared, 7", and the cube of the scmi- Boajor axis a of the ellipse. 60 62 Looking Out: Rockets, Satellites, Space Travel Example. Calculate the height of a satellite in a 24-hr orbit about the earth if it has been observed that a satellite at a mean distance of 4,100 mi from the center of the earth has a period of 5,000 sec. From Kepler's third law 3Y 7V a.* we wish to find o 2 when T~ = 1 day = 8.6 X 10 4 see = (I/V a , = [tf)l 4,100 mi = 27,000 mi a-i Kepler's lawn apply to the ideal ease of only two bodies mov- ing under their mutual gravitational attraction. But in space travel, effects of other bodies have to he considered. To consider the feasibility of certain proposals or devices, one starts by examining qualitative orbits. Such trajectories are predicted with the aid of simplifying assumptions: that the moon moves in a circle around the earth, that the earth may be con- sidered symmetrical, that any disturbing masses are in the orbit plane of the space vehicle, etc. The precision trajectories needed for actual space travel do not allow these approximations. Hence the calculations become enormously more complicated.* 6.2 NEWTON'S DERIVATION OF KEPLER'S LAWS As a test of his theory of universal gravitation, Newton desired to show that Kepler's laws could be derived from the law of gravi- tation and he desired to investigate the more general problem: What kind of motion is necessary according to that law? In its basic statement, the law of universal gravitation applies only to particles ("point" masses). Newton needed first to show that the attraction for an exterior particle exerted by a spherical mass (either homogeneous or somewhat like the earth, made up of concentric homogeneous shells) was directly proportional to the total mass of the sphere and inversely proportional to the square of the distance of the particle from the sphere's center. Newton's difficulty in establishing this principle to his satisfaction may have been the cause of his delaying some twenty years in publishing his conclusions. * Precision rocket orbits are discussed in S. Herrick, "Astrody- namies," I). Van Xostrand Company, Inc., Princeton, N'..l., 1959. Motion of Bodies in Space 63 The orbits of all the planets (except Pluto) are very nearly circles, with the sun at the common center. Kepler's third law can be derived by equating the centripetal force to the gravita- tional force (Sec. u.3) to obtain T* 4» ! i* = m = con8tant Kepler's second law, the law of equal areas, follows whenever the interaction between two particles is in the direction of the line joining them. The force need not follow an inverse square. Fig. 6.3 Derivation of Kepler's second law. Let Pi, P->, and P% be points along a planet's orbit marking the position of the planet at time intervals of 1 sec. Then the distance PiPz is numerically equal to the planet's velocity 1% and /V J 3 is numerically the velocity r., in the next second. When the only force acting on the planet is in the direction of the sun, this force has a component zero perpendicular to line / J 2 ,S'. Hence the com- ponent of the planet's velocity perpendicular to / J -,.S' must be unchanged, according to Newton's first law of motion: r tL = u 2l . The area swept during the first second by the line joining the planet and the sun is &P1P1. The area swept in the next second is SP t P 3 . These triangles have the same base P°S and equal alti- tudes v ± ; hence they have equal areas. The derivation of Kepler's first law is more lengthy, and it involves differential equations.* The question is: Given an in- verse-square law of attraction, what shape must a planet's (or comet's) orbit have? The answer turns out to be: The orbit will be one of the conic sections with the attracting body (sun) in one focus, * A derivation without calculus is presented in Jay Orear, "Funda- mental Physics," pp. 70-73, .John Wiley & Sons, Inc., New York, 19GI. 64 Looking Out: Rockets. Satellites, Space Travel Conic sections are curves? obtained by taking plane slicas of a solid circular cone (Fig. G.4). The cone sliced parallel to its base (I) gives a circle. If the cut is slanted, the section is an ellipse (2). With greater slant, the section is a parabola (3). With still greater slant, the section is a hyperbola (4). 1. Circle 2. Ellipse 3. Parabola (parallel to lineaO) 4. Hyperbola Fig. 6.4 Basic orbits related to conic sections. Fig. 6.5 Newton's proposal for an earth satellite, [(a) From Sir Isaac Newton, "Mafnemoficai Principles . . . ," edited by F. Co/oW, University of California Press, Berkeley, Calif., 1934. (b) From E. M. Rogers, "Physics for the inquiring Mind," Princeton University Press, Princeton, N./., 1960.1 Motion of Bodies in Space 65 The significance of the various conic-section trajectories may be clarified by an example based on Newton's own suggestion for an earth satellite. About 1660 he predicted in a drawing (Fig, (5.o) that if a cannon ball could be fired with a muzzle velocity of "i mi/sec, it would circle the earth as shown. The Sputnik and Explorer satellites did achieve this velocity, For a low-flying earth satellite in a nearly circular orbit, equating the centripetal acceleration r/r to g gives v = \/gr — 9.8 X (>.' r > X 10 6 m/sec = 8 km/sec or 5 mi /sec Now alter Newton's drawing (Fig. (U>) by considering the mass of the earth to be concentrated at point E (Fig. (i.(i). Con- Hyperbola Fig. 6,6 Conic orbits cotangent at satellite launching point p. ^ider that a satellite is to be launched at point p with a velocity Perpendicular to the line K v . J-et the, circle represent the orbit of the satellite described in the preceding paragraph. The effect of the earth's attraction is to cause the satellite to fall r = Igl 1 = 4.9 m toward the earth in the same second it travels 8 km along the tangent. The two displacements bring the satellite back to the 66 Looking Out: Rockets, Satellites, Space Travel same distance it had before. So, during each second, the satellite falls toward the earth but never gets any closer. Now suppose that the satellite's velocity is made less than S km/sec. The earth's effect of 4.0 m each second is unchanged. So the satellite will fall closer to the earth along the smaller ellipse of Pig. 6.0. Since the earth is not a "point" as implied in Fig. 0.0, the satellite actually will not he able to complete the elliptical orbit but rather will strike the earth after traveling a trajectory which is si portion of an ellipse (Fig. 6.5). (The smaller ellipse of Fig. 0.0 could represent the path of a comet or planet about the sun at A'.) If the satellite at p were given a velocity somewhat greater than 8 km/sec, the 4.9 m by which it would fall to the earth each second would be insufficient to hold the distance constant. The satellite would climb away from the earth on the larger of the two ellipses (Fig. 0.0). With decreasing speed the satellite would arrive at a point a opposite the start. There the centrifugal reaction would be insufficient to overcome gravitational attrac- tion, though the latter would also have decreased. Accordingly, the satellite would begin to fall back toward the earth, regaining speed along the elliptic path until it reached point j> with the same velocity as at the start. Increasing the satellite's velocity at p still more would semi it off along the parabola shown in Fig. 6.6. Still greater velocities would carry the satellite away from the earth along a hyperbolic path. In either case the attraction of the earth would be insuffi- cient to decrease the radial velocity of the satellite enough to cause it to return. QUESTIONS AND PROBLEMS 1. The periods of revolution of the planets Mercury, Venus, Mars, and Jupiter are, respectively. 0.241, 0.017, 1.88, and 11.9 years. Find Iheir mean distances from the sins, expressed in astronomical units (1 a.u. = distance from sun 1o earth). 2. .Jupiter lias a radius of 74,000 km. A satellite completes an orbit, about Jupiter every J 6.7 days. The radius of the orbil of the satellite is 27 times the radius of the planet. Compute the mass of Jupiter. 3. What docs Kepler's second law say about the duration of winter in the Southern Hemisphere (which occurs in .inly when the earth is farthest from the sun) compared with winter in the Northern Hemisphere? 4. Show the correctness of Kepler's third law of planetary motion Motion of Bodies in Space 67 by equating the centripetal force required to keep a planet in its (circu- lar) orbit to the gravitational force due to the sun's attraction. 5. What is the mass of a planet, .1/,., if it is observed to have a moon revolving about it at a distance /?, center to center, in period T? 6. If the earth, considered to be spherical, were to shrink to 0.9 of its present radius, what changes would occur (a) in the length of the solar day, (/>) in the value of g at the North Pole, (<•) in the value of g at the equator? 7. The earth satellite Kxplorer III had a highly eccentric orbit witli perigee at a height of 109 mi. At this point the velocity was 27,600 ft/sec in a direction perpendicular to the radius to the center of the earth. Show that this speed is too great for a circular orbit at the radius (R, + h) of 4,109 mi. Hence the satellite described an elliptical orbit. Its apogee was at the height 1,630 mi. Show that the speed at apogee was too small for a circular orbit at radius 5,630 mi. It is the supreme art of the teacher to awaken joy in creative expression and knowledge. A. Einstein, motto for the Astronomy Building, Pasadena Junior College Putting on the spectacles of science in expectation of finding the answer to everything looked at signifies inner blindness. J, Frank Dobie Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. H. Poincare Science is organized knowledge. Herbert Spencer Science is nothing but perception. Plato Travel to Moon and Planets We first throw a little something into the skies, then a little more, then a shipload of instruments then our- selves. . Fritz Zwicky The solar system, consisting of 9 planets moving in elliptical paths around the sun, 31 known moons, and many other bodies all in motion, does not invite simple straight-line travel from the earth to a selected destination. To conserve both power and time, departure dates and trajectories must be chosen which utilize favorable positions and relative velocities. Conditions favorable for return passage may not occur until some time later. Owing to the ever-changing distribution of bodies in the solar system, no two courses between even the same two bodies are likely to be the same. The calculations of desired trajectories and corrections of the course while in flight are complex tasks for computing machines. 7.1 INVITATION TO INTERPLANETARY FLIGHT Despite obvious difficulties of travel in the solar system, there are some interesting favorable factors. The space between the earth 68 Travel to Moon and Planets 69 and other bodies in the solar system is almost a perfect vacuum. This is an ideal environment for a space vehicle to move at speeds which make it practical to travel interplanetary distances. Since I he earth is one of the smaller planets, it requires a comparatively low escape velocity. Its relatively thin atmosphere offers less resistance to rapidly ascending and descending objects. The fa el that the planets lie in nearly the same plane and move in the same sense makes it possible for an interplanetary traveler to apply the orbital speed of one planet in launching himself to another. The fact that the elliptical orbits of the planets are nearly circular means that the energy requirements to transfer a spaceship from one orbit to another do not vary greatly for different points of departure along the orbit. Finally, most planets rotate in the same direction in which they revolve about the sun. So a space- ship launched at the surface of one of these planets can get an added push by taking oil" in the direction of rotation. 7.2 LAUNCHING Before it is launched, the space vehicle is at the earth's distance from the sun, and it is moving with the earth's speed around the sun (about 100,000 ft/sec). If launched at greater than the earth's escape velocity, the vehicle will take up an independent orbit around the sun, at a speed somewhat different from that of the earth. Fig. 7.1 Launching! to inner and to outer planets. 70 Looking Out: Rockets, Satellites, Space Travel If it is launched in the same direction as the earth's orbital motion, the vehicle will have a speed greater than that of the earth (l'"ig. 7,1.1), and could reach the outer planets, Mai's, Jupiter, etc., if properly directed. The minimum starting speeds required to reach these planets arc given in Table 7.1. Table 7.1 Minimum launching speeds, with transit times to reach the planets Plonet Minimum launching speed, ftsee Transit time Mercury 44,000 1 10 days Venus 38,000 150 days Mars 38,000 260 days Jupiter 46,000 2.7 years Saturn 49,000 6 years Uranus 51,000 1 6 yea rs Neptune 52,000 31 years Pluto 53,000 46 years From R. W. Buchheim, "Space Handbook," 1958. If the vehicle is launched "backward," against the earth's velocity, it will move in an orbit like H in Fig. 7.1, so it could reach VettUS or Mercury. However, it requires almost, as much energy to propel a vehicle in to Mercury as to propel it out to Jupiter. 7.3 ROCKET GUIDANCE In the flight of an unmanned probe, satellite, or missile, one or more boosters provide the initial impulse, but after burnout the remainder of the flight is unpowered. The vehicle coasts in the complex gravitational field of interplanetary space. The accuracy of guidance is generally determined by the position and velocity at the instant free flight begins. Figure 7.2 gives an idea of the maximum allowable errors of angular alignment, and vehicle velocity at power cutoff for several kinds of moon-directed mis-ions. Inertiul Isiyro) or radio-guidance techniques are ade- quate for such relatively simple missions. Interplanetary expeditious present complex problems of guid- ance. First, a launching site might be chosen at not more than 2:}° north or south latitude. This is the angle of inclination of the ecliptic plane to the earth's equator. The vehicle would be launched into a satellite orbit around the earth, in the ecliptic Travel to Moon and Planets 71 S i.o § > 'G 1 _o « > o a, 1= I 0.1 | S. .8 o.oi E § o> 0.001 Scientific sotellire (±100 mi) Impact on moon (+100 Around moon return to braking eclipse 0.001 0.01 0.1 Speed error, per cent Fig. 7.2 Maximum permissible errors for alignment of velocity vector and for speed at power cutoff. (Genera' Electric, "Space Facts.") Initio! ascent -^ ^~v •'/- Earth ' ^^ Fig. 7.3 Possible flight paths to Mars, plane, |>'ig. 7.H. With the vehicle in orbit, an ion-propulsion sys- tem might be started to cause the vehicle to spiral out into a legion where the sun's gravitational field is stronger than the 72 Looking Out: Rackets, Satellites, Space Travel earth's. The vehicle would then be guided into an elliptical trans- fer orbit around the sun, planned to intercept the orbit of the, destination planet. Where these orbits intersect, the vehicle would be directed into an orbit around the destination planet. Radio or inertial guidance techniques could serve in the early stages of such an interplanetary flight, but would probably be inadequate for interplanetary missions of a year or more in duration, A three-dimensional form of present-day two-dimen- sional celestial navigation may be necessary. A useful instrument for establishing a reference direction is the horizon seeker which senses the infrared radiation of the earth or some other warm body. Optical trackers and magnetometers may also provide data to establish the vehicle's position. To orient and stabilize a space vehicle, torque is produced, either by the ejection of mass (rocket exhaust) or by the rotation of a mass within the vehicle. The internal type of torque control serves to rotate the vehicle about its center of mass ; it does not influence the flight path. 7.4 RETURN THROUGH THE ATMOSPHERE To return safely to earth, a space vehicle must overcome the problems of penetrating the earth's atmosphere. There are three general types of reentry path, each with its characteristic de- celeration pattern: direct descent, orbit decay, and lifting descent. These are illustrated in Kig. 7.4. In direct descent into the atmos- phere, the maximum deceleration experienced is independent of the drag characteristics of the vehicle, but depends on the path angle, initial velocity, and characteristics of the atmosphere. The altitude at which maximum deceleration occurs does depend on the drag characteristics of the vehicle. For entry of the atmosphere in orbit decay, the vehicle exe- cutes many revolutions about the earth in a very gradual spiral that becomes more and more nearly circular. The rate of energy loss through aerodynamic drag is sufficiently small so that the vehicle's kinetic and potential energies adjust to a momentary "equilibrium" orbit, with potential energy decreasing and kinetic energy increasing. Thus the velocity of the vehicle actually in- creases in the start of orbit decay. The final phase of descent is similar to that of direct descent at a shallow angle. In a lifting descent, the aerodynamic characteristics of the Travel to Moon and Planets 73 vehicle are used to obtain a very gradual penetration of the atmosphere. The path angle is generally small, a few tenths of a degree, and is adjusted to the forces acting on the vehicle. Decel- eration increases gradually and can be limited to a relatively small value. The more gradual the descent, the longer is the time required and the longer is the range. Starting at a given altitude and velocity, a direct descent may traverse a distance of a few hun- dred miles and require about A min. An orbit, decay might cover a range of several thousand miles and require 5 to 10 min. A Ballistic rocket Direct from space Fig. 7.4 Different types of atmospheric entry. (Generaf Elec- tric, "Space Fads") lifting descent from the same point might range over o,000 to 10,000 mi and require 2 hi*. When a vehicle penetrates the atmosphere, the reduction of the vehicle's energy is accompanied by an increase in the thermal energy of the surrounding air, some of which is communicated to the surface of the vehicle. At very high altitude, about one-half the energy loss appears as heat in the body. At lower altitudes, the heating is produced not directly at the vehicle's surface but in the air between the shock wave and the vehicle. Heat is trans- ferred from the hot gases of this region to the vehicle by conduc- tion, convection, and radiation. 7 -5 THE NEXT DECADE OF SPACE RESEARCH Historically, man's attempts to predict the future of science and technology have shown a tendency to be overly optimistic about 74 Looking Out: Rockets, Satellites, Space Travel what will bo accomplished in the immediate future and too con- servative about the long-range future. It has been predicted that man's curiosity about the unknowns of outer space can be only partly satisfied by the placing of meters in outer space; eventually he will want to go there to see for himself. But it is probable that the extent and pace of space research in the foreseeable future will be determined by what are regarded as our military requirements. The military advantage to be gained from putting man in space is at least debatable, From the standpoint of psychological warfare, there may be better ways of demonstrating our scientific prowess. For man's future happiness, more important pure- science experiments might be performed in other fields, such, as medicine. Yet many dedicated scientists feel that man-in-space experiments are important to our chances of survival This viewpoint is stated by Dr. Simon Ramo in the following terms. Suppose two rival nations base their security on a race for wisdom in the use of limited technical and physical resources. Suppose, however, that the first nation makes one decision in contrast to that of the second: It decides that man will never be needed in space. These two nations start to develop their weapons systems of the future. One group has maximum flexibility; the other has some prohibitions. "To achieve this maximum of flexibility, it is very clear to me that the United States must prepare for putting man in space," says Dr. Ramo. On the assumption, then, that we shall have military- sponsored programs in space technology, one can make some predictions for the near future. Many projects involving com- munications, weather prediction, manned satellite stations, and exploration within the solar system will probably be fulfilled. Exploration beyond the solar system now seems unattainable chiefly from considerations of time and power. Man's life is short when compared with the time required to reach the nearest star, even in a vehicle traveling with a speed approaching that of light. The other problem, "Where is the energy In he obtained for long voyages or to lift large masses into space?" may find an answer in the achievement of a nuclear-fusion reactor. It is the thermo- nuclear bomb which threatens to make the earth a very unpleas- ant place. Ironically, the energy of a controlled thermonuclear reaction may provide us with the power resources for a migration into space. PART Looking In: Atomic and Nuclear Physics 8 The Atomic Idea Science does not know its debt to imagi- nation, R. W. Emerson Although Democritus had introduced the word "atom," it was the English school teacher John Dal ton (1803) who made fertile the assumption that matter is not divisible indefinitely but rather is composed of ultimate particles called atoms. Physics dealing with phenomena on a scale large enough to be visible to the un- aided eye was well understood by the year 1890. Then a remark- able mutation occurred in science, caused by the series of dis- coveries made in the decade 1895-1905: 1895 Discovery of x rays by Roentgen 18% Discovery of radioactivity by lleequerel 1897 Identification of the electron by Thomson H'OO Statement of the basic postulate of quantum theory by Planck Pi05 Formulation of the theory of relativity by Einstein It became clear that the structure of matter was much more complicated than had previously been thought. The term "modern physics" is often used to designate micro- scopic (atomic and nuclear) physics, investigated from the view- 77 78 Looking In: Atomic and Nuclear Physics point of quantum theory and relativity, as distinct from the macroscopic or "classical" physics which was known before 1890. 8.1 DEVELOPMENT OF THE ATOMIC CONCEPT OF MATTER The existence of atoms has been inferred from many experiments, the earliest of which were studies of simple chemical professes. By 1800, some 30 elements had been identified and the formation of chemical compounds had been studied. Lavoisier showed that mass appeared to be conserved in chemical reactions. Proust, Dalton, Berzelius, and Richter discovered "laws" which may be summarized in the statements: 1. A particular compound always contains the same elements chemically united in the same proportions by weight. (Law of definite proportions.) 2. When two elements A and B combine as constituents of more than one compound, the weights of B which unite with a fixed weight of A (and vice versa) aj-e related to each other as the ratios of whole numbers, which are usually small. (Law of multiple proportions.) Dalton showed that these chemical laws could be explained most directly in terms of an atomic theory of matter. Its assump- tions arer 1. All matter is made up of elementary particles (atoms) which retain their identity in chemical reactions. 2. The atoms of any pure substance (element) are alike (on the average, at least) in mass and other physical properties, :>. Atoms combine, in simple numerical proportions, to form com- pounds. Dalton's clear formulation of the atomic concept of matter is the first important landmark in the development of modern atomic physics. 8.2 AVOGADRO'S NUMBER Joseph Gay-Lussac (1808) showed that, at a constant tempera- ture and pressure, gases combine in simple ratios by volume. Amadeo Avogadro was led (181 1) to make the important assump- tion that equal volumes of different gases under the same coudi- The Atomic Idea 79 tions of temperature and pressure contain the same number of molecules. This hypothesis guided Bcrzelius and others in deter- mining the ratio of combining weights (e.g., is water HO or H s O?). A molecule is the smallest particle of any substance (element or compound) as it normally exists. An atom is the smallest portion of an element found in a mole- cule of any of its compounds. An atom is the smallest portion of an element that can enter into chemical combination. By measuring combining weights, it is possible to determine the relative masses of atoms of various elements. We may arrange them in order of increasing mass, assigning a number to each to indicate its relative mass. Since only the ratios of the numbers are important, we may assign one number arbitrarily to a particular atom and adjust the others accordingly. Conventionally, the number Hi (exactly) was assigned to an oxygen atom. Then by Avogadro's hypothesis, for any gaseous substance .Molecular mass of substance density of substance density of oxygen (Os) X 32.000 (8.1) Since 19(51, the Commission on Symbols, Unite and Nomen- clature in Physics has defined the atomic mass unit as one-twelfth of the mass of an atom of the carbon- 1 "2 nuclide. The number representing the mass of any atom on this scale is called its atomic mass. (The term "atomic weight" is also used.) On this scale, which differs only very slightly from the former one, the mass of the hydrogen atom is nearly 1 amu and the heaviest known atom has a mass of about 250 amu. We shall use the symbol .1* to represent, as needed, either atomic mass or molecular mass. A quantity of any substance whose mass, in grams, is numeri- cally equal to its molecular mass is called a mole. In the mks system we define the "kilogram mole" as: I kmole of a substance is that quantity whose mass in kilograms is numerically equal to its atomic (or molecular) mass. The mass of 1 kmole of any sub- stance is thus .1 * kg. The volume occupied by a mole of any gas is called a gram molecular volume. It is 22.4 liters for a gas at 0°C and 76 cm of mercury pressure. The numerical value of Avogadro's number is not easy to 80 Looking In: Atomic and Nuclear Physics The Atomic Idea 81 measure, and it was not known for some time after Avogadro's hypothesis was accepted. This constant can be determined inde- pendently from experiments in electrolysis, Brownian motion, radioactivity, and x-ray diffraction in crystals. The currently accepted value of Avogadro's number is N A = (6.02486 -f-O.OOOHi) X 10 2S molcculcs/kmolc (8.2) Example. Compote the number of atoms in a 1.5-mg sample of lead, atomic mass 207. The mass of 207 atomic muss units (amu) may be thought of as 207 kg/mote. Then 1 5 X I0~ 6 ksr N = 207ki7kmole X 6 ° 25 X ^" ^m./kmole = 4.36 X I0 1S atoms 8.3 THE IDEAL GAS LAW The gaseous state of matter is simplest to analyze, chiefly because the molecules of a gas arc far apart and do not exert appreciable forces on each other. The behavior of a gas is expressed by an equation of state, which relates pressure, temperature, and volume when the gas is in equilibrium. Numerous empirical equa- tions have been suggested to describe the behavior of gases. The simplest is pV - nRT (8.3) where p = pressure of gas V = volume T = absolute temperature n = number of moles (or kilomoles) of gas present In mks units the proportionality constant R is called the gas con- stant per kilomole, and from experiment, it has the value R = 8.317 X 10 s joules/(kmole){°K) (8.4) When other units are used for the variables in Eq. (8.3), the gas constant will be expressed differently; for example, R = 1.987 cat/ (mole) (°K) = 0.082();> liter-atm/(mole)(°K) = 8.317 X 10' ergs/ (mole) (°K) Xo actual gas obeys Eq. (8.3) precisely at any nonzero pres- sure. Hut this equation holds for all gases when the pressure is reduced sufficiently. For then the molecules occupy an insignifi- cant fraction of the volume of the container and the widely separated molecules exert no attracting forces on each other. It is from these considerations that Eq. (8.3) has importance as the "ideal-gas" equation of state. 8.4 KINETIC THEORY OF GASES Kinetic theory treats atomic and molecular processes and reaction rates by applying elementary methods of mechanics and statis- tics. We shall examine what the kinetic theory has to say about the observed properties of a gas: its pressure, volume, and temperature. y\ /\ / Fig. 8.1 Model for kinetic theory of gas pressure. West i i i i i i i — East A / We shall consider a gas confined at a fixed temperature in a cubical container with each side of length L. We make the fol- lowing assumptions: I. The molecules have negligible volume; they are "points." The molecules move in random directions, but every molecule has the same speed v (obviously an oversimplification, which we shall reconsider soon). The molecules exert forces only in collisions. The collisions with the walls are clastic. The number of molecules is very large, justifying use of statistics. The pressure of the gas may be calculated as the force per unit area at a wall. Let .V he the total number of molecules in a cubical container (Tig. 8.1). Then N/8 will be bouncing lietween the east wall and the west wall. Each molecule in this group 82 Looking In: Atomic and Nuclear Physics strikes the east wall v/2L times per second. In each clastic impact the velocity of the molecule changes sign and the change in its momentum is mv — ( — me) = 2mv, where m is the mass of the molecule. From Newton's second law, the force on the cast wall is the total momentum change per second at that wall „ N i> ,. Nmu % The pressure is given by P = A = J_ Nmjfi _ 1 Nmv* l- a£ "a v (8.5) where V = I* is the volume of the container. To compare this prediction with the ideal-gas equation we may rewrite Eq. (8.5) as pV = i(AW) = \8%m* - %NB t (8.6) where E k is the translational kinetic energy of one molecule. Combining Imjs. (8.3) and (8.0), we have nRT = fJVA't (87) suggesting that the absolute temperature of a gas is proportional to the kinetic energy of its molecules. Further, since N/n is the number of molecules per kilomole, that is, Avogadro's number #,1, we have for the kinetic energy of a molecule 3 R h\ = - — T = : 'I:T (8.8) The constant k, called Boltzmann's constant, is the gas constant per molecule k =■ — = 8-317 X 10 a j oule s/(kmole )(°K') A', 6.025 X 1 M molecules/ kmole = 1.38 X 10" M joule/ (molecule) (°IC) (8.9) An improvement can be made in our simples) statement of the kinetic theory by removing the second assumption above, that all molecules have the same speed. Instead we can say that for any particular molecule v- = v t * + vf + v?. If we have a large The Atomic Idea 83 number of molecules moving at random, the average values of iv, tv, and rr are all equal. Equation (8.6) then becomes pV = llVmi* ami the v of our earlier discussion is replaced by the "rms ve- locity," the square root of the mean value of the square of the velocity, VP. 8,5 DISTRIBUTION OF MOLECULAR SPEEDS In a gas at a given temperature and pressure, we expect that some molecules will have speeds in excess of the root-mean-square value, and others will have smaller speeds. Clerk Maxwell applied the laws of probability to find the distribution of speeds in a large number of molecules in a kinetic- theory gas. He obtained the result * X 4 \Ztt (&f« e - mT =/s*r f}„ (8.10) where N = total number of molecules N r dv/N - fraction of all molecules whose speeds are between v and v + dv T = absolute temperature fe = Boltzmann constant m = mass of a molecule For a gas at any given temperature, the number of molecules in a speed interval A» increases up to a maximum at the most probable speed v p of Fig. 8.2a and then decreases toward zero at high speeds. The distribution is not symmetrical about »„, for the lowest speed is zero, but the theory predicts no upper limit for the speed a molecule can attain. For this reason, the average value v of all speeds is somewhat larger than «„. The root-mean-square value v rim is still larger. As the temperature of a gas is increased, the most probable speed of the molecules increases in accord with the meaning of temperature (Eq. 8.7). The range of speeds is greater (Fig. 8.2b); * L. U. Loeb, "Kinetic Theory of Gases," McGraw-Hill Book Com- pany, Inc., New York, 1934; Leigh Page, "Introduction to Theoretical Physics," chap. 9, D. Van Nostrand Co., Inc., Princeton, K.J., 1935. 84 Looking In: Atomic and Nuclear Physics ~kT /itr No. (a) *-/Fs /»=m 200 400 600 800 m/sec Fig. 8.2 (o) Maxwell distribution of molecular speeds at 0°C. (b) Maxwell speed distributions at three different temper- atures, T 3 > T-i > T|. there is an increase iti (lie number of molecules which have speeds greater than a given speed. 8.6 MEAN FREE PATH TIxe mean free patli is defined as the average distance a molecule travels between collisions with other molecules. Assume that each molecule is a sphere of radius R. Consider the motion of a par- ticular molecule among all the other molecules of a gas. It will lilt- any molecule whose center lies within a cylinder of radius 2J9 around its path (Fig. &.',t). In going a distance L, the molecule sweeps out a volume ir(2R)' 1 L. If there are >i molecules per unit volume in the gas, the moving molecule will bitx(2R) i Ln molc- The Atomic Idea 85 rules in going a distance L. Its mean free path X is the average distance per collision: X = r(2R)*Ln 4*Rhi (8.11) This equation is based on the picture of the single moving molecule hitting other molecules which are stationary. Actually the molecule hits moving targets. The collision frequency is 2R m Fig. 8.3 Path of a molecule. increased as a result. More complete analysis shows that the nieaii free path is reduced to X = I 4tt y/2 tr-n (8.12) Example. I" helium gas at 0°C and I atm pressure, the mean free path of one molecule (or atom, He) is 1.86 X 10 -7 m. Estimate the radius of 8 helium atom. The number of molecules per cubic meter under standard condi- tions m n = .V,, R = 22.4 (m 3 . bnole) 1 = 2.(59 X 10" molceulcs/m' {Aw y/2 Xh) j = 1.05 X I0-"m which agrees in order of magnitude with other methods of measure- ment. Note that the mean free path X is about 1.86 X 10 7 m/2 86 Looking In: Atomic and Nuclear Physics (1.05 X 10 u m) = 900 moleeuliir diameters. From Bq. (S.8) the speed of the He atoms is u = = — = 1,310 m/sec 2 m So the frequency of collision is v 1,310 m/sec X 1.86 X 10-' m/(2 X 1.05 X 10 10 m) or 7 billion collisions per Becond. = 7 X lO'sec - ' 8.7 MEASUREMENT OF MOLECULAR SPEEDS An experimental verification of the. distribution of molecular speeds predicted by kinetic theory was reported l>y Stern in (a) 1 m G&rtn (4) ^~=©=" Fig. 8.4 Apparatus for measuring molecular speeds, (a) Stem's rotating drum, (b) Lammert's slotted disks. i926. Atoms (Ilg) from an oven at known temperature pass through a slit <S and enter a cylinder C through a narrow slit, in its wiill (Fiji. 8.4a). With the cylinder stationary, the molecular beam reaches O diametrically opposite the entrance slit. Hut if the cylinder is rotated rapidly, the molecular beam is interrupted. If a point on (he cylinder wall rotates clockwise from O to () v in the time it takes an Tig atom to cross the diameter d, then the trace left on the wall by IIg atoms will be displaced counter- clockwise a distance 00% (= 0\0) from the reference point. The speed of the atoms can be calculated from Speed of atoms diameter d Speed of drum surface displacement O a O The Atomic Idea 87 In Lammert's method, two disks each with ")0 notches were mounted fi cm apart on a rotating axis (Fig. 8.46), in an arrange- ment similar to that used by I'izeau to measure the speed of light. For a particular constant speed of rotation of the disks, only those Hg atoms of a certain speed will be able to pass through both notched wheels and reach the collector plate P. !3y varying the speed of rotation and by determining the number of atoms received at P as a function of their speed, one obtains results in N(v) u.o 0,?0 ' - -\ - -1 0.15 0.10 ^■"1 0.05 90 140 190 240 290 Speed, m/sec 340 390 Fig. 8.5 Speed distribution of mercury vapor molecules at 100°C. (6. tammert, Zeitschrift her Physik, 56(3-4): 244-253 (1929).] good agreement with Maxwell's predicted distribution (Fig. •S..V). (The slight discrepancies with the predicted values, shown dotted, were attributed to difficulties of alignment.) 8.8 SPECIFIC HEATS Consider a gas confined at constant volume which is heated. The specific heat C? of the gas is defined as the heat required to raise the temperature of a unit quantity of the gas one degree. This heat is stored in the form of increased kinetic energy of the gas molecules. From Bq, (8.8) the increase in the internal energy of 1 mole divided by the increase in temperature is given by (8.13) 88 Looking In: Atomic and Nuclear Physics Thus our basic kinetic theory makes the challenging prediction that all gases have the same value of specific heat C\ - §[1.987 calV(moIe)(°C)| = 2.98 cal/(mole)(°C) The value predicted checks well with experimental data for monatomic gases, but not for gases whose molecules are made up of two or more atoms. Toble 8.1 Specific heats (at 15°C) Type of gas Gas C,. (experimental), cal/{moleH°C) Monatomic He 2.98 A 3.00 Diatomic Hi 4.80 o, 4.96 N, 4.94 CO 4.95 Polyatomic C0 2 6.74 NH 6.78 C^Hg 9.50 To explain the data of Table 8.1, we may ask whether heating may result in energy being stored in forms other than transla- tional kinetic energy of molecules, expressible in terms such as ■frnvf. In a dumbbell-model diatomic molecule (Fig. 8.6b), there may be kinetic energy of rotation, expressible in terms such as ^/ur. If the two atoms can vibrate and have a force constant k, there will be vibrational energy expressible as \l.\r-. Each independent mode of absorbing energy is called a degree of freedom,/. A theorem of cquipartition of energy, stated by Max- well, says that for a large number of particles which obey New- tonian mechanics, the available energy is equally divided among the degrees of freedom, .]/.V for each. Thus modified, our kinetic theory eau be made to agree fairly well with experimental data for monatomic gases (/ = 8, C„ = 35/2) and for diatomic gases (/ = 5, C = 5B/2). One finds * The calorie was originally defined as the amount of heat necessary to raise the temperature of one gram of water one centigrade degree In 1948 it was redefined as 1 caloric = 4.1840 joules. The large calorie {kcal or Cal) used in nutritional measurements is 1,000 times as large. The Atomic Idea 89 experimentally (1) that, contrary to kinetic theory, C T varies with temperature, and (2) that for polyatomic molecules we need to devise empirical models that differ from gas to gas. We have come to the limit of validity of classical mechanics when we seek to describe the behavior of very small particles of matter (molecules and atoms). Quantum theory is the extension of classical theory which we need for this {Chaps, I o and Hi). We shall reach another Fig. 8.6 Degrees of free- dom: independent modes of energy absorption. (a) O (b) Degrees of freedom 3: trans!. 3 tronsl, 2 rota . 3 trans) . 2 rota . 1 vibra. (c) OmKD (d) limitation in Newtonian mechanics when we deal with particles which are moving very fast (» — * c). Relativity (Chap. 1-1) modi- fies Newtonian mechanics in this case. PROBLEMS 1. Copper which has a specific gravity S.9 has an atomic mass R3.8 amu. What is the average volume per atom of copper? Ans. 1.2 X 10" cm 3 2. Compute the rms speed at 0°C of the molecules of («) CO», (b) H 2 , (c) Xo. Aits. H02 in sec, 1X4 m '.-ee. !()2 m 'sec 3. If the average distance between collisions of CO* molecules under I atm pressure and at Q°C is ft. 29 X 10 6 in, what, is the time between collisions? Ans. 1.6 X 10"" sec 90 Looking In: Atomic and Nuclear Physics ■i. In a certain electron microscope, electrons travel 1.0 m from electron gun to screen. To avoid scattering of electrons by residual molecules of nitrogen in I lie vacuum chamber, below what pressure would you recommend operating (lie microscope? The radius of a nitrogen atom is about 2 X 10 10 m. Ana. p < 8 X 10~ s atm 5. What pressure will 10 gin of helium exert if contained in a 50-cm* cylinder at 2l°C-? Would a cylinder rated at 100 atm maximum safe pressure be safe to hold this helium? 4ns. p « 1,000 atm Equipped with his five senses, man explores the universe around him and calls the adventure Science. E. P. Hubble A series of judgments, revised without ceasing, goes to make up the incon- testable progress of science. DuClaux The main difference of modern scientific research from that of the middle ages lies in its collective character, in the fact that every fruitful experiment is published, every new discovery of relationship explained . . . Scientific research is a triumph over natural instinct, over that mean instinct which makes a man keep knowledge to himself and use it slyly to his own advan- tage ... To science this is a crime. H. G. Wells Wave Motion To the mathematician the problems of wave motion offer a field for his highest power of analysis; to the physicist they suggest experiments demanding all the skill at his disposal ; to the engineer and to those who go down to the sea in ships these prob- lems are matters of life and death, while to the poet and the artist they are "the sea dancing to its own music " Henry Crew In the study of wave motion wc arc concerned with the propaga- tion of disturbances in physical systems. A wave is a description of a disturbance wliicli propagates from one point in a medium to other points, without causing atiy permanent displacement in the medium as a whole. Tints sound is a type of wave motion; wind is not. Wave motion occurs in a medium in which energy can be stored in both kinetic and potential form. In an elastic material, kinetic energy results from inertia and is stored in the motion of 91 92 Looking In: Atomic and Nuclear Physics the molecules, whereas potential energy results from the displace- ment of molecules against an elastic restoring force. In an electro- magnetic wave, we may regard kinetic energy as stored in the magnetic field and potential energy in the electric field. In a traveling wave, one part of the medium disturbs an adjacent part so that kinetic energy at one point is transferred into poten- tial energy at an adjacent one, and that potential energy becomes kinetic energy at still another point, and so on. 9.1 TYPES OF WAVES A wave is a disturbance that moves through a medium in such a manner that at any point the displacement is a function of time, while at any instant the displacement at a point is a function of the position of the point. The medium as a whole does not pro- gress in the direction of motion of the wave. Waves are usually described mathematically in terms of their amplitude (maximum displacement from equilibrium) and how the displacement varies with both space and time. This requires solution of the wave equation consistent with the boundary conditions for the particu- lar case being studied. In cases most often considered, the wave equation is a second-order, linear, partial differential equation. The general solutions of the wave equation for a one-dimensional space coordinate x are of the form * = F(x - vt) + G(x + vt) (0.1) The functions P and (! are determined by the boundary condi- tions, and the speed v by the properties of the medium. The first term represents a wave traveling in the positive x direction; the second term represents a wave traveling in the negative x direc- tion. These are usually sine or cosine waves, for the one-dimen- sional case. A wave moving on a string is an example of a one-dimensional wave. Hippies on water are two-dimensional waves. Acoustic and light waves are three-dimensional. Waves may be classified in accordance with the motion of individual particles. Transverse waves and longitudinal waves are the most common types, but there are others. For example, as a wave moves on the surface of water, the path followed by an individual particle is either a circle or an ellipse. Wave Motion 93 Elastic waves, of which acoustic or sound waves are a particu- lar kind, require a medium having two properties, elasticity and inertia. Elasticity of the medium is needed to provide a force to restore a displaced particle fo its original position. Inertia is needed to enable the displaced particle to transfer momentum to a neighboring particle. In an elastic medium one may have, in addition to a longitudinal or a transverse wave, a shear wave. This is a rotational wave which causes an element of the medium to change its shape without a change of volume. Light waves, radio waves, and other electromagnetic waves are not elastic waves and therefore can travel in free space as well as in transparent media. In a vacuum all electromagnetic waves travel with constant speed, approximately :j X 10* m/sec. -km Tvv n I » . . (a) (6) Fig. 9.1 Wave fronts; (a) plane; (b) spherical. Arrows represent rays. In material media the speed is less, and its value depends on the medium. Waves may be classified further in terms of time: the perio- dicity or lack of periodicity of the disturbance. If a long coil spring ("Slinky") is stretched out on a table, a single sidewise movement at one end will send a pulse or single wave along the spring. Kaeh particle remains at rest until the puke reaches it, then moves for a short time, and returns to rest, However, a continuing to-and-fro motion applied to the end of the spring will produce a train of waves. If the motion is periodic, we shall have a periodic train of waves. An important special case of periodic wave is a simple harmonic wave in which each particle is given an acceleration proportional to its displacement and directed toward the equilibrium position. 94 Looking In: Atomic and Nuclear Physics An aid in visualizing waves is the idea of a wave front. A wave front is a surface drawn through points undergoing a similar disturbance at a given instant. The location of a disturbance (pulse) at successive e(|iial time intervals may be indicated by drawing successive wave fronts. A line perpendicular to a wave front, showing the direction of motion of the wave, is called a ray. Wave fronts spreading from a point source in a homogeneous medium are spherical. Hut at large distance from the source a section of the wave front may be treated as practically plane. 9.2 FUNDAMENTAL RELATIONS A wave is commonly identified in terms of either its wavelength \ or its frequency J". In any kind of wave motion these two quanti- ses are related to the velocity of propagation > by the simple equation /A = v (9.2) The period is the reciprocal of the frequency. The amplitude A Wave Motion 95 Medium t — "-Pi -A* Medium 2 (f 2 >u,) (a) -A (6) Wavelength ■* — — m- Wavelength \j ■* — Fig. 9.2 When a wave passes from one medium to another, in which the wave speed is different, (a) the frequency is constant, (fa) the wavelength changes. is the maximum value attained by the variable of the wave (e.g., the displacement) at a given point in space. The frequency of a wave remains constant under all circum- stances except for a relative motion between the source of the wave and the observer (see Sec. 9.15, Doppler Effect). The speed of propagation, however, is dependent on the properties of the medium (and, sometimes, also on the frequency). Hence the wavelength will vary with speed in accordance with Eq. (9.2), as suggested in Fig, 9.2. 9.3 WAVE FORM A wave form is a pictorial representation of a wave obtained by plotting the displacement with respect to lime or distance. When a wave is traveling along a string in the x direction, the shape of the string at some instant I = can be expressed by an equation V ■ /(#) when t = (9,3) which states that the transverse displacement :/ is some function f(x) of the distance x along the string. If the wave is moving to the right with a speed v, the equation of the wave at some later time I is y = f{x - Bt) (9.4) This gives the same wave shape about the point x = vl at time J as we observed about point x = at time t = 0. The relative positioti (displacement) of two points in a wave is called the phase. Two points which have displacements of the ( = t=*t a b \e d( <vt Fig, 9.3 A traveling wave. same magnitude and sign (a and h in Fig. 9.3) arc said to have the same phase, or to be "in phase." Points c andrf do not have the same phase, for although both have zero displacement, the dis- placement is decreasing at c, increasing at d. To follow a particu- 96 Looking In: Atomic and Nuclear Physics lar phase in an ongoing wave, wc ask how x changes with t when x — vt has some particular constant phase value P. Differentia- tion of x — vt = P gives dx/dl = v. So v is the phase velocity of the wave. A wave form of considerable importance is one defined by a sine function y = f/o sin -r- (■*•" — i>0 = ij» si" 2» A (x t) (9.5) The maximum displacement //« is called the amplitude of the wave. The wavelength A represents the distance between two points which have the same phase, [■'or a given (, the displacement // is the same at x, at x + X, at x + 2X, etc. The period T is the time required for the wave to travel a distance of one wavelength X, so X = vT, From the second form of LCq. (9,5) it is apparent that y has the same value at the times t, t + T, i + 2T, etc., at a given position x. Wave Motion 97 - • t T 2 T Sine wave litt y=y sin jr Square wave «= 1,3,5- Saw-tooth wave J'0 J'„ r-> 1 tint it = 1,2,3- Fig. 9.4 Some wove forms, defined in terms of sine functions of the frequency, t T, and its multiples, n/T. There are many wave forms of interest in physics. To specify a particular wave Form, one chooses the appropriate function /(.!■)• It is possible to represent any periodic wave form mathe- matically as a Fourier scries of sine and cosine terms at har- monics (multiples) of the frequency 1, T. Examples are shown in Fig. 9.4. 9.4 WAVES IN A LIQUID The waves which we most often see arc those which occur on Ihe surface of a body of liquid. Waves also occur within a liquid. Their propagation is made use of in marine equipment such as the fathometer and Sofar. Fig. 9.5 Liquid in a channel, showing two positions of o liquid element being considered. A quantitative description of a wave motion often can be obtained by applying fundamental laws of physics to a particular situation. As an example, consider a channel of unit width and vertical walls which contains a layer of liquid whose initial depth is outlined by the solid line in Fig. 9.5. Examine how this can move to successive positions, as suggested by the dotted line. We shall apply two physical laws: (1) No liquid disappears or is created during the process. 2 The rate of increase <.n momentum of any liquid clement must be equal to the net external force applied to that element. The force applied to a fluid element at a cross section such as .1.1 equals the area times the average pres- sure. Since we have assumed unit width for the channel, area = I X It. The element considered will be the liquid which initially lies under the solid line of Fig. 9.5, between .4.1 and ('('. After time I, this same liquid will be under the dotted line between .I'.l' and 98 Looking In: Atomic and Nuclear Physics C'C. The distance from .1.1 to A' A' is u^l, where h, is the speed of the liquid lying to tfie left of B. To satisfy the first requirement mentioned above (At — hi)x = kiUtt (9.6) where x is the distance the liquid originally at rest lias been accel- erated during time interval J. To satisfy the second requirement f (A, 2 - A.') = "*f -' (9.7) where ipgki is the average pressure (above atmospheric) in the liquid of depth Ai, \pgh-i is the average pressure at CC where the depth is As, and p is the mass per unit volume of liquid. The left- hand side of Eq. (9.7) is the net horizontal force on the element of liquid considered. The mass of liquid ph«x is accelerated from rest to speed a 5 in time t, so (ph«x/t)ui is the rate of change of momentum. The two requirements expressed by Eqs. (9.6) and (9.7) now give for the speed m, of the particles of liquid , „ (Ai - A;) 2 (Ai + h*)g 2hih a (9.8) But the wave speed x/t at which the front of the deeper layer advances is given by v= i = 4 (At + A 2 )flig 2A 3 (9.9) The wave speed v is greater than the speed u of the material particles. In this simplified treatment, we have disregarded energy )o->i>. variation of speed with depth, and a detailed specification of the shape of the wave front. Vet we have obtained a valid description of the tidal bores which occur in certain rivers. Such a surge wave is sometimes employed as a means of dissipating flow energy at the bottom of a dam spillway. If the channel is so designed that i/ t = ~r then the velocity of the surge relative to earth is zero. This form of surge is known us a hydraulic jump. it can often lie viewed on a small scale by allowing water to flow- Wave Motion 99 from a faucet into a basin. The flow can be adjusted so that in the basin there can be seen an inner zone consisting of a thin layer of water moving rapidly outward. Surrounding this is an outer zone which is a thicker, more slowly moving layer. The manner in which a continuing oscillatory wave is propa- gated may be examined from considerations similar to those just suggested for a .-urge wave. In shallow water (say, h ,\ < ,'„ . a 2(, Fig. 9.6 A large-amplitude wove steepens to form a bore (Fig. 9.5). wave of small amplitude will be propagated without change of shape at a speed y/gh, which is consistent with Eq. (9.9), If, however, the wave height is an appreciable fraction of the liquid depth, the wave speed is significantly greater at positions of greater depth. The wave front becomes successively steeper (Fig. 9.(i), and a bore starts to form. In deep-water waves, individual fluid particles move in approximately circular orbits (Fig. 9.7). At the surface, the radius StiM-woter leve * — "^ Shallow water Bottom (b) Fig. 9.7 Orbital motion of fluid particles for surface waves (a) in deep water, and (b) in shallow water. of the orbit of a particle is equal to the amplitude of the wave. Hut the radius decreases exponentially with depth, and a region of almost zero particle motion is soon reached ; hence the behavior of the wave is unaffected by the total depth of the liquid. lu shallow water there can tie im vortical motion of particles at the bottom. The orbits of the particles are ellipses in which the vertical axis becomes zero at the bottom (Fig. 9.76). 100 Looking in: Atomic and Nuclear Physics A wave lias equal amounts of potential energy, owing to particle displacement above or below the still-water level, and kinetic energy, owing to the motion of the particles in their orbits. The speed at which energy is transmitted in the direction of wave travel is called the group velocity n. us distinct from the phase velocity v = x/l. In deep-water waves the group velocity is one- half the phase velocity. In shallow-water waves u = e. 9.5 SOUND WAVES IN A GAS In sound waves usually encountered, the intensity is so small that the changes in temperature and pressure in the wave are a very small fraction of the ambient temperature and pressure. Plane wave Front «1 Pi «2 Pi Fig. 9.8 Plane wave front in a gas. These waves l ravel at a speed which depends only on the ambient state of the fluid. The propagation of a sound wave in three dimensions can be derived from fundamental physical principles starting in this way. Imagine, a small prism or a packet of gas enclosed by a weightless dcformable membrane. The mass within this packet remains constant. The elasticity is expressed by the ideal-gas law. The inertia appears in Newton's second law, from which the equation for the wave propagation can be derived. A simpler procedure may be followed in describing the special case of a plane wave front moving from right to left at constant speed it, in a gas initially at rest and having density pi. To an observer moving with this wave front there will appear to be a steady flow of gas from left to rigbl across the wave Ironi (Kg, 9.8), Wave Motion 101 Since in a steady flow there can be no accumulation of the mass at the wave front, Pi*'i = wis (9.10) where p« is the density of gas at the right of the wave front and h 2 is the velocity of this gas relative to the observer moving with the wave front. Also, an increase in gas momentum across the wave front requires a drop in pressure from pi to p 2 : P2>1? — Pl»l 2 = Pl - P'J (9.11) Tiiis expression is obviously related to Bernoulli's theorem for the steady flow of on incompressible fluid (p = const). If we consider (he fluid flowing past two different cross sections of a pipe at different elevations A] and h~ and apply the principle of conservation of energy, we get (9.12) «{As - Ai) + \ («*• - »,*) = p, - pj Bernoulli's theorem thus says (hal at any two points along a streamline in an ideal fluid in steady flow, the sum of the pressure, I he potential energy per unit volume, and the kinetic energy per unit volume have the same value. For a small disturbance where the fractional changes in gas velocity, density, and pressure are much smaller than unity, these changes across the wave front can be written as it* = u\ + du, Pz = p\ + dp, and p : = pi + dp. When we substitute these in Kqs. (9.10) and (t).ll) and neglect product terms of differential quantities, we have Pl du + tt r dp - (9.10a) 2p,u, du + urdp = -dp (9.1 In) By eliminating du from these two equations, we obtain an expres- sion for the wave speed u - F p (9.13) Laplace assumed the compressions and expansions associated with sound waves should obey the adiabatic gas law, pp—> = constant where y is the ratio of the specific heats, C p /C r . If this Relationship for p and p is assumed, the speed of sound becomes tti = VyRT = J?l (<ut) 102 Looking In: Atomic and Nuclear Physics This result, based on the adiabatic law, does not hold for liquids, for gases at extreme pressures and temperatures, or for acoustic waves of very high frequencies. However, the pressure fluctua- tions in sound waves range from about 10 9 to 10~ 3 atm, which justifies the asumption of small disturbance in deriving Bq. (9.14). 9.6 SHOCK WAVES IN A GAS In a wave of large amplitude, the wave speed is higher than wi in regions of condensation (p > pi) and lower than »i in regions of rarefaction. This causes the wave to distort as it propagates. Regions of higher condensation overtake those of lower condensa- tion (Fig. 9.9). The thin "characteristic lines" are shown for cor- Fig. 9.9 When wave speed increases with wave amplitude, the wave form becomes distorted at successive time intervals, 1„, I,, 2f i. f D = responding points in the wave. The slope di/dx of these lines is inversely proportional to the speed. The net effect is to steepen compression regions and to flatten expansion regions. Before the situation represented at 2d is reached, friction and heat-transfer effects counteract the steepening tendency. The compression part of the wave propagates without further distortion. It is then a shock wave. Bomb blasts start as shock waves, large-amplitude compres- sion waves. Planes traveling at speeds greater than the speed of sound (Much niiniher = speed of body/ local speed of sound > 1) generate shock waves which are responsible for the sonic boom sometimes heard and felt on the ground. When an astronaut reenters the earth's atmosphere, the early motion of his vehicle is determined by its shock wave and can be estimated from the size and velocty of the vehicle and the known temperature, pressure, and density relations for the wave. Wave Motion 103 9.7 WAVES IN SOLIDS Different types of acoustic waves may occur in solids, depending on the way in which potential energy is stored in the solid. Transverse waves on flexible stretched strings are described by an equation of the form 3P" 6\r a (9. If)) where ;/ is the displacement of the string at a point ,r. The speed of propagation v is equal to the square root of the ratio of the tension to the mass per unit length of the string: * = 4 In (9.10) Acoustic waves occur in bars when the bar i> brm and re- leased. Here the restoring force is due to the moment of the forces about the neutral plane in the bar and depends on the cross- sectional dimensions and on Young's modulus. Seismic waves which travel through the ground originate from natural readjustment of the faults in the earth's crust or from explosions. Both body and surface waves result. The body waves, which travel through the interior of the earth, may be classified into dilationai (longitudinal) waves, which are similar to acoustic waves in compressible fluids, and shear (transverse) waves, which occur on account of the large shear modulus of most elastic solids. From known relationships between propagation speeds and the mechanical properties of various substances, seismologists obtain from seismograms valuable information about the strnc- i lire of the earth. Such information can be applied to prospecting Tor mines and wells. 9.8 ELECTROMAGNETIC WAVES ■lames C. Maxwell recognized about 180-1 that the basic equations for electric and magnetic lields could be combined to give an equation which resembled the wave equation for mechanical waves in a fluid {see Sec. Fi.fi). 104 Looking In; Atomic and Nuclear Physics 9.9 SUPERPOSITION OF WAVES For many kinds of waves, two or more waves ean pass through the same space independently of one another. One can distinguish the notes of a particular instrument while listening to a full orchestra. The displacement of a particle in the medium at any instant is just the sum of the displacements it would be given by each wave independently. The principle of superposition states that the net displacement of a particle is the vector sum of the displacements the individual waves alone would give it. This principle holds for an elastic medium whenever the restoring force is proportional to the deformation. Superposition holds for Fig. 9.10 Analysis of a complex wave form. electromagnetic waves because of the linear relations between electric and magnetic fields, The superposition principle does not hold in every ease. It fails when the equations describing the wave motions are not linear. An acoustic shock wave has a quadratic wave equation; superposition does not hold. Hippies which can cross gentle ocean swells cannot preserve their identity in breakers. Intermodulation distortion occurs in an electronic amplifier when the system fails to combine two tones linearly. An important consequence of the superposition principle is that it provides a means of analyzing a complicated wave motion as a combination of simple waves. Joseph Fourier showed that any smooth periodic function may be represented as the sum of a number of sine and cosine functions having frequencies which are multiples of a single basic frequency. The displacement of a Wave Motion 105 particle in the medium transmitting a complex wave is given by an equation of the form of y = A i sin co( + Ai sin 2ut + At sin Swi + ■ - • + B« + if i cos oil + Bi cos 2U + B,i cos :io>t + (9.17) In Fig. 9.10, the wave (dotted) which has an approximately square wave form is shown to be equivalent to three component waves with frequencies in the ratio 1 :3:o and amplitudes in the ratio 1 :| : g. The Fourier series representing the square wave is A A . y = A sin id + tt sin JW + ■=■ sm out + 9.10 INTENSITY OF A WAVE (9.18) In any wave, energy is transmitted through the medium in the direction in which the wave travels. The amplitude of the wave, which is the amplitude of vibration of the particles in the medium, is related to the transmission of energy. Each particle has energy of vibration which it passes on to the succeeding particles. In simple harmonic motion, where there is no damping, the energy of a vibrating particle changes from kinetic to potential and back, the total energy remaining constant. We may find this constant energy from an expression for the maximum kinetic energy E k - lm(u ui:ix y = ( 2 r ')' " \m@rfv<>y 2ir % mf 2 y^ (9.19) where y<> = amplitude of vibration T — period / = frequency m — mass of the particle The energy per unit volume in the medium is the energy per parti- cle times the number n of particles per unit volume ~ - a2* s 8*/W - -iTr-pp-ih? (9.20) where p = mn is the density. The intensity / of a wave is defined as the energy transferred per unit time per unit area normal to the direction of motion of 106 Looking In: Atomic and Nuclear Physics the wave. The energy tluil travels through such an area per unit time is that contained in a volume which has unit cross section and a length equal numerically to the speed 8 of the wave. From Eq. (9.20) / = 2Teh>ppij,r (9.21) The intensity is directly proportional to the square of the ampli- tude and to (he square of the frequency of the wave. When u wave originates at a point source and travels outward through a uniform medium, at some instant the energy is passing through the surface of a sphere. A moment later the same energy is passing through a larger spherical surface. Since the total energy per unit time is the same at the two surfaces, the intensity is inversely proportional to the area 4?rr 2 of the surface: / = lirr* /.": (9.22) If instead we have a line source (e.g., a fluorescent lamp), the energy is spread over successively larger cylindrical surfaces. The intensity is inversely proportional to the area 27rr/ of the cylindrical surface: f = 2jrrt hi r (9.23) Here the intensity is inversely proportional to the distance r. For a plane source (e.g., a skylight), which is large compared to the distance from the source, the energy passes through suc- cessive planes of equal area. There is no divergence of the rays. In this case the intensity is independent of distance. As a wave passes through any medium, some energy is ab- sorbed by the medium. Hence the energy |>;i-<iim through suc- cessive surfaces decreases faster than expected from the change in area alone. The decrease in intensity due to absorption of energy is called damping, A wave whose amplitude decreases for this reason is called a damped wave. 9.11 INTERFERENCE OF WAVES The physical effect of superposing two or more wave motions is called interference. Where waves arrive in phase, the interfer- Wave Motion 107 ence is constructive. The amplitude is the sum of the amplitudes of the individual waves. Where waves arrive 180° or X/2 out of phase, the interference 1 is destructive. When two wave trains of different frequency interfere, a series of alternate maxim;! and minima is produced in the amplitude of the vibration (l-'ig. i).ll). The frequency of these "beats" is the difference of the two wave frequencies. A familiar example occurs in sound. If two tones of slightly different frequency are sounded together, one perceives that the loudness pulsates at the beat wwvwm i 1 1 1 1 1 1 1 wwwvwm i 1 1 1 1 1 1 1 1 Fig. 9.11 Two waves of different frequency combine to couse beats. Two coincidences per unit time are shown for wave trains of frequencies 10 and 12. Frequency. Thus if the tones are middle O (2(i!/sec) and O sharp (280.5/scc), there will be 16.5 beats sec. 9.12 DIFFRACTION The bending of a wave around an obstacle is called diffraction. Diffraction is readily observed as ripples on water bend around a stick placed in their path. The principles of diffraction and interference are applied in the measurement of wavelength of light with an optical diffraction grating. A transmission grating is a glass plate upon which is ruled many equally spaced lines, usually several thousands per centimeter. A parallel beam of monochromatic light falling normal to this grating (l-'ig. 9.12) sends waves in all forward directions from each slit. Along certain definite directions waves from adjacent slits are in phase and reinforce each other. Consider parallel rays making an angle with OB, the normal to the grating, which are brought to focus at a point P by an 108 Looking In: Atomic and Nuclear Physics achromatic Ions, If ray AP travels a distance X farther than ray CP, then waves from .1 and C will interfere constructively at /' for they differ in phase by a whole number of wavelength*. The wave front CD makes an angle 8 with the grating. From the small- est right triangle, the path difference X is seen to be CA sin 0. The distance ('A between corresponding points in the ruling is called the grating space />. The condition for reinforcement in the direction 8 is b sin = X (first order) (9.24) There are other directions on each side of OH for which waves from adjacent slits differ by 2X, 3X, -IX, etc., and for which the Fig. 9.12 Diffraction grating. corresponding bright images P>, /\, etc., are called the second- order, third-order, etc., images. The grating equation in more general form is b sin 6 = A T X (9.2.-)) where N is the order of the spectrum and b is the grating space. When white light fulls on the grating, it is dispersed into its component colors. Spectra are produced at Pi, r\ etc. The dis- persion is greater in the higher-order spectra. In each, the colors appear in the sequence violet (small X) to red (large X) with in- creasing deviation. Example. A yellow line and a blue tine of the mercury-arc spectrum h;ivc wavelengths of 5,791 A and 4,358 A, respectively. In the spectrum Wave Motion 109 formed by a grating that has 5.000 lines/in., compute the separation of these two lines in the third-order spectrum o = ttIsL cm = 5.08 X 10" « cm o.OOO 5,791 A = 5.791 X 10" r ' am 4^68 A = 4.358 X 10 * am sinfl = ^ = 3(5^?1_X 10 = <n.O ... „. e _ 2();r " b 5.08 X 10-' cm ' sin 8i, = Separation 3(4,358 X IP" 6 cm) 5.08 X 10- 4 cm '* — ft ~ 5,3 = 0.258 ft, = 15.0° 9.13 STANDING WAVES Tf a wave on reaching the boundary of a medium is totally reflected., the reflected wave proceeds in the opposite direction ■fW (a) (b) Fig. 9.13 Standing waves from superposition of waves traveling in opposite directions; R is the resultant of A and S. The envelope of a standing wave is shown in (M. and with equal amplitude (big. 9.13.!. The incident and reflected waves add according to the principle of superposition. Two such waves, proceeding to the right and left, may be represented by the equations MM) The resultant may be written (9.26) i/ x = yo sin y-i = !/ B sin : ,'/t + t/s — tf » s i n 110 Looking In: Atomic and Nuclear Physics We may use the trigonometric relation for the sum of the sines of two angles sin A + sin II = 2 sin l(A + B) cos %{A - B) to put Eq. (9.26) in the form 2wx „ I y = 2y sin -y- cos 2w ■=-, (9.27) This is the equation for a standing (no n progressing) wave. A particle at a particular point % executes simple harmonic motion. All particles vihrate with the same frequency. But the amplitude is not the same for all particles; the amplitude varies with the location x. The points x = «X/2 (where n is an integer), at which sin (2tx, X) = 0, show no displacement and arc called nodes. The amplitude has a maximum value 2i/ n at points .<■ = 2« + I X and 2 2' such points are called autinodos, or loops (Fig. 9.136). In general, when a wave reaches a boundary, there is partial reflection and partial transmission. Consider a stretched string attached to a second string. When a wave in the first string reaches the boundary joining the strings, the. reflected wave has smaller amplitude than that of the incident wave because the transmitted wave in the second siring carries away some of the, incident energy. If the second string has a smaller linear density than the first, reflection occurs without change of phase. If the second string has a greater linear density than the first, there is a phase shift of 180° on reflection. From Eq. (9.2fi), it is evident that the wave travels more slowly in the denser string. From the relation X - v/f, we conclude that in the denser string the wave- length is shorter. In a study of light waves we frequently observe this phenomenon of change of speed and wavelength as light passes from one; medium to another. 9.14 RESONANCE Free or natural oscillation refers to the oscillation of a body or a system which has been given a displacement from equilibrium and then is not acted on by any external or driving force. The body or system will generally have several distinct frequencies of natural oscillation. Wave Motion til If a system which can oscillate is acted upon by periodic impulses having a frequency equal or nearly equal to the natural frequencies of the system, oscillations will occur with relatively large amplitude. This vigorous response of a system to pulses nearly synchronous with one of its natural frequencies is called resonance. Let us determine the natural frequencies of a stretched string. When standing waves are established in the string, the end points will be nodes. There may be other nodes in between. So the wavelength of the standing waves can have many distinct values. Since the distance between adjacent nodes is X/2, in a string of Vib rotor Fig. 9.14 Standing waves in a string driven at a frequency nearly equal to a natural frequency, length t there must be exactly an integral number n of half wavelengths, X 2, so X = * n 7i = 1,2,3, From Kqs. (9.1) and (9.2ti), the natural frequencies of vibration are } 21 yitn/t n= 1,2,3, (9.28) These relations may be demonstrated in a string one end of which receives energy from a vibrator, such as an electrically driven tuning fork. The string passes over a pulley, I' in Fig. i>. 14, and is attached to a weight which maintains the string under tension I<\ The frequency / of the wave is that of the vibrator. The wavelength is 9 1 i / / V F_ in I 112 Looking In: Atomic and Nuclear Physics The wavelength may be varied by changing the tension F, which changes the wave speed v. Whenever the wavelength becomes nearly equal to 2l/n, standing waves of large amplitude may be observed. The string is then vibrating in one of its natural modes and is in resonance with the vibrator. Example, What forte must be exerted on the string, using the apparatus of Kg. 9.14, to produce resonance with the string vibrating in one loop? The vibration has a frequency 20/sec, the string has a length 18 ft and weighs o.O ok. From Eq. (9.28), with n = I, F = 4 Ifm = 4(18 ft) 400 6.0 slug sec 2 10 X 32 3-1 11) 9.15 DOPPLER EFFECT There is a change in the observed frequency of sound, light, or other waves caused by motion of the source or of the observer, \ i ami liar example is the increase in pitch of a train whistle as the train approaches and a decrease in pitch as the train passes. In the radar system used for traffic control, the speed of a car is estimated from the Doppler frequency shift in the radar beam reflected from the car. In acoustics the Doppler effect deals with cases of relative mo- tion between listener and source, plus the effect of any motion of the medium. If the source moves toward a stationary observer with speed vn, waves emitted with a frequency f s appear to have their wavelength shortened in the ratio (u — Vn)/u, because of the crowding of the waves in the direction of motion of the source (big. 9.15), Theses waves, however, arrive at the listener with the speed u characteristic of the medium. If, instead, the listener moves with speed u L toward a sta- tionary source, the waves appear to him to arrive with speed m + v,l- The wavelength in this case is the same as that measured when both listener and source are at rest in the medium. Xow consider motion of the medium. Let Vm be the component of its velocity taken positive in direction from listener to source. The velocity components v f , and vs are taken to be positive in the direction from listener to source. Wave Motion 113 Then the general equation relating the observed frequency fi and the source frequency fs is h f* u + v L — 9u u + vs — I'M (9.29) There an; important differences between the acoustical and the optical Doppler effects. ( I ) The optical frequency change does Fig, 9.15 Doppler effect due to motion of the source S toward observer O. Wave front 1 was emitted when the source was at position 1; wave front 2 Was emitted when the source was at position 2, etc. The drawing shows positions of Wave fronts when the source is at S. not depend on whether it is the source or the observer that is moving with respect to the other. (2) An optical frequency change is observed when the source (or observer) moves at right angles to the line connecting source and observer. No acoustical frequency shift is observed in the corresponding case. (3) Motion of the medium through which light waves are propagated docs not affect the observed frequency. Analysis of the Doppler effect for electromagnetic waves 114 Looking In: Atomic and Nuclear Physics (light) requires use of Lorcntz transformations and the relativity postulate that the wave speed c is the same as measured by all observers. The result for the observed frequency fa is Vl - WJ&) j0 Js 1 - (u/c) cos 6„ . cos B s + (v/c) COS do = . , / ■- } -■; — a I + {V/C) COS Or (0.30) (9.31) where 0<, is the angle measured in the observer frame and 9s is the angle that would be measured in the source frame if it were moving with velocity v relative to the observer frame. The term transverse Dtrppler effect refers to the relativistic, direction-independent factor in the equations above, fo m Is \/l — (u'/c 1 )- This shows that the observed frequency will be less than the source frequency regardless of the apparent direction of motion of the source. The radial Doppler effect is the direction-dependent factor and, like the acoustical Doppler effect, is understandable on the basis of classical physics, fo « /s/[l — if>/c) cos 0«|. In MK-18 II. ]•'„ Ives and (1. It. Stilwcll measured frequencies in the spectrum emitted by moving hydrogen atoms and compared the frequency shifts with those predicted by the equations above for the transverse Doppler effect. This became an experimental verification of the special theory of relativity and of the "dilata- tion of time" (Sec. 14.!)). QUESTIONS AND PROBLEMS !. Knergv can lie transferred by particles as well as by waves. How can you distinguish experimentally between these methods of energy transfer? 2. When waves interfere, is there a loss of energy? Kxplain. 3. Why don't wo observe interference effects between the light beams omitted from two flashlights, or between the sound waves from violins in an orchestra? 4. A line source (fluorescent lamp) emits a cylindrical expanding wave. Assuming the medium absorbs no energy, find how the amplitude and intensity of the wave depend on the distance from the source. 5. A cord 75 cm long has a mass of 0.252 gm. It is stretched by a load of 2.0 kg. What is the speed of a transverse wave in this cord? Am. 242 m/scc Wave Motion 115 6. find the speed of a compressional wave in a steel rail whoso density is 490 lb/ ft 3 and for which Young's modulus has a value 29 X 10 6 lb/in.*. Am. 5,200 ft/scc 7. Compute the speed of sound waves in air at 0°C. The average molecular weight of air is 29, y = 1. 40. and R = 8.3 X 10 a joules/ (kmole)CK). 8. If a person inhales hydrogen and then speaks, how will the characteristics of his voice be changed? How would the situation be changed if carbon dioxide were used? 9. A student places a small sodium vapor lamp just in front of a blackboard. Standing 20.0 ft away, he views the light at right angle-; to (be blackboard while holding in front of his eye a transmission grating ruled with 14,500 lines, in. He has his assistant mark on the hoard the positions of the first -order diffracted images on each side of the lamp. The distance between these marks is found to he 14 ft, 2 in. Compute the wavelength of the light. An ocean traveler has even more vividly the impression that the ocean is made of waves than that it is made of water. A. S. Eddington False facts are highly injurious to the progress of science, for they often endure long; but false views, if supported by some evidence, do little harm, for every one takes a salutary pleasure in proving their falseness. C. R. Darwin What art was to the ancient world, science is to the modern. Benjamin Disraeli Science and art belong to the whole world, and the barriers of nationality vanish before them. Goethe -* Electric and Magnetic Forces 117 10 Electric and Magnetic Forces 1 cannot help thinking while I dwell upon them that this discovery of magnet-electricity (induction) is thB greatest experimental result ever obtained by an in- vestigator, J. Tyndall Electric charges and electric and magnetic forces are important in many experiments? designed to reveal the structure and be- havior of atoms. All visual information cornea to us in electro- magnetic waves, and study of the ultimate structure of atomic nuclei depends on electromagnetic processes and detectors. We shall outline here only the main ideas in electricity and magne- tism needed for our study of atomic and nuclear physics. The study of electricity dates from the observation ((500 ux.) that bits of straw and other materials arc attracted to rubbed amber. The study of magnetism dates back at least as far, to the observation that magnetite stones attract iron (but not other substances, lieneralh , These two sciences were developed sepa- rately until 1820, when Hans Christian Oersted observed a rela- tion between them: An electric current in a wire can affect a magnetic compass needle. However, the fact that electricity and 116 magnetism were initially developed as separate sciences has led to some inconveniences in concepts and units which the viewpoint of the inks units (which we shall use) seeks to minimize. 10.1 CHARGE AND MATTER We anticipate experimental evidence described in later chapters to summarize, some modern basic knowledge. Experiments on the electrification or charging of bodies show that there are two Nucleons: Neutrons O O Protons © * Nuclei Atoms :h©(§)i Mo I ecu Electrons 9© '"(©0©) Compounds I ( }p ) - — ** le Visible matter Fig. 10.1 Composition of matter. kinds of charge. A glass rod may be rubbed with silk, placed in a stirrup, and suspended horizontally on a silk thread. If a second ulass rod is also rubbed with silk and then brought near the rubbed end of the first rod, the two rods will repel each other. Hut a hard-rubber rod electrified by rubbing with fur will attract the glass ra cJ. Two rubber rods rubbed with fur will repel each other. The charges on the glass and hard rubber must be different. We add the following details to the atomic picture of Chap. 8. An atom has most of its mass concentrated in a very tiny (10 -13 cm) nucleus. The simplest atom, hydrogen, has a nucleus which comprises a single proton. All other nuclei contain, in addition to protons, one or more neutrons. Each atom has circulating 118 Looking In: Atomic and Nuclear Physics around its nucleus a number of electrons equal to the number of protons within the nucleus. The mass of the electron is about 1/1,840 the mass of a proton or neutron (Table 10.1). An arbi- trary convention adopted in Franklin's time for the sign of the two kinds of electric charge leads us to call the electron charge negative, the proton charge positive. A neutron lias zero charge. Toble 10.1 Properties of some baste particles Particle Symbol Charge Mass, kg Electron Proton Neutron P n -e (= -5.60 X 10-"coul] 9.108 X 10- 31 1.672 X 10"" 1.675 X 10 « An element may be designated by symbols zEl- 1 , such as «Be* for berillium. The atomic: number Z represents the number of pro- tons (or electrons) in the atom. Its mass number A represents the number of nucleons (neutrons and protons) in the nucleus. The number of neutrons is .1 — Z, The chemical properties of an atom are determined by its atomic number. Two atoms which have the same atomic number, but whose nuclei contain different numbers of neutrons, are said to be isotopes of the given element. Objects can be electrified, or charged, either positively or negatively by the removal or addition of electrons. Charges of like sign repel ; unlike charges attract. In the atomic model proposed by Niels Bohr in 1913, elec- trons arc pictured as whirling about the nucleus in circular or elliptical orbits. The centripetal force needed To hold an elect nm in its orbit is provided by the force of attraction exerted by the positive nucleus on the negative electron. In addition to the electrostatic (coulomb) forces between charges, there are forces which depend on the relative motion of the charges. These forces determine the magnetic behavior of matter. 10.2 COULOMB'S LAW Coulomb's law (1785) expresses the experimental observation that the force of attraction (or repulsion) exerted by one charged Electric and Magnetic Forces 119 object on another is proportional to the product of the charges, q x and q 2 , and inversely proportional to the square of the distance r between them (where the objects are regarded as "point" masses) : F = k <?><7s (10.1) The proportionality constant k is a positive number whose value depends on the system of units. Tn the electrostatic system of units (esu), the unit of charge is defined conveniently to make k = 1 in Eq. (10.1): One stat- coulomb is that quantity of charge which repels a like charge with a force of one dyne when the charges are spaced one centi- meter apart in a vacuum (or practically, in air). However, the meter-kilogram-second (mks) system of units defines a unit for current (ampere) as a fundamental unit; the unit for charge (coulomb) becomes a derived unit. The ampere is defined in terms of an electromagnetic experiment. The ampere is the strength of that constant current which, maintained in two parallel, straight, and very long conductors of negligible cross section placed in a vacuum at a distance of one meter from each other, produces between these conductors a force of 2 X 10 "' newtou per meter of their length. The coulomb is that charge transferred by an unvarying current of one ampere in one second. In principle, we have only to measure the force, in newtous, between two 1-coul charges separated by 1 m in vacuum to hud k in mks units. The experimental value is k = 8.987 X 10° « 9 X 10* newton-m-'/coul 2 (10.2) In the so-called rationalized mks system of units, a different constant t , called the permittivity of free space, is introduced in the equation for Coulomb's law /■' - so that eo = 4ttc r 2 (10.3) 1 4irfc 4tt(8.987 X JO 11 ) newton-mVcoul 2 = 8.85 X 10 '- coul7newton-m 2 (10.4) 120 Looking In: Atomic and Nuclear Physics The arbitrary inclusion of the Factor 4ir in Coulomb's fundamental law makes certain derived formulas more convenient. No it's then appear in formulas referring to plane surfaces, a factor 2tt appears in "cylindrical" formulas, and 4jt appears in formulas relating to spherical symmetry. For example, Table 10.2 gives expressions for the capacitance C (charge held per unit potential difference) for capacitors of different symmetry as expressed in unratioual- ized units and in rationalized mks units. The vehemence with which questions of units have long been argued may be inferred from Oliver Ileaviside's statement (1893) Table 10.2 Comparison of expressions for capacitonce Unrationalized units ■ i </ — ^ Plane capacitor C = Awd Rationalized units C = iAJd L i i| Coaxial cylinders ji (I 2 In b;a C= 2 In b/a Concentric spheres C= — Ah b -a „_ i (4r)ab b -a that "the unnatural suppression of Aw in the funrationalizedj formula for central force, where it has the right to he, drives it into the blood, there to multiply itself, and afterward to break out all over the body of electromagnetic theory." The mks units were adopted by international agreement for scientific and engineering use beginning in 1940, but actual acceptance of the mks system has progressed slowly. We shall use the rationalized mks system of units. The statement that Coulomb's law applies to "point" charges means, practically, that charges qy and q-> must be associated with bodies whose dimensions are negligibly small compared to r. The evaluation of the constant k above holds only for the case where the two charged particles are in vacuum. If they are immersed in some medium, say, oil, the polarization of its molecules greatly Electric and Magnetic Forces 121 diminishes the force between charges r/i and q*. Coulomb's law is then written F — <7ifj a /4jrer s , where « is replaced by the larger number e, the permittivity of the material in question. 10.3 ELECTRIC FIELD INTENSITY If there are several charges Q u Q->, Q 3 , . . . , in fixed positions, and we bring up another charge 7, it will experience a force. We say that the fixed charges set up an electrostatic field about them and the charge q experiences a force when in this field. We define the electric field intensity as the net force per unit + charge R force charge +q (10.3) Electric field intensity is a vector quantity. Its mks units are newtons per coulomb (or volts per meter, from Sec. 10.4). We can often calculate the value of K at each point of a region of space; these values determine the force on (and hence the motion of) a charged particle in that region. F = gE (10.(>) The electric field near an isolated point charge Q is easily calculated. If a test charge q is brought to a distance r from Q, it experiences a coulomb force F = q ® ~ (4x eo )r 2 The magnitude of the field is then P _ Q K - +q (4areo)r* (10.7) (10.8) This electric field intensity is represented by a vector which, at each point in space, points directly away from Q if Q is positive or directly toward Q if Q is negative. Example. Two charges, q, = —75 X 10 -9 coul and q t = + 75 X 10 -9 t'tiLil, are 8.0 cm apart in uir. Find I he electric field intensity E at a 122 Looking In: Atomic and Nuclear Physics point P, which is 5.0 era from each charge (Fig. 10.2). The field intensity due to 171 U represented by the vector PA and is given by 2?, _-«L 4weor i = 75 X 10-'(9 .0 X 10') newton (0.05)* i-oul = 27 X 10* newton/coul The field Ei due to charge q* is also 27 X 10* newton/coul in magnitude, but its direction is that represented by vector PC. The resultant field E 9 l =-75xT0" 8 ?,=> +75xicr D Fig. 10.2 is represented by vector PR. Since triangles TKiP and PRC are similar, one may write the following proportion: PB = DG or _B 8.0 cm PC PG 27 X 10* newton/coul 5.0 cm E = 43 X 10* newton/coul, parallel to the tine joining q t and q it 10.4 ELECTRIC POTENTIAL In electrical phenomena the concept that is important in cases of energy transfer is that of potential difference. If we move a charge through an electric field, we exert a force through a distance, and so do work. The force exerted at each point is proportional to the amount of charge moved, and thus the total work is proportional in this charge. The electric potential difference between positions b and a is the work done per unit + charge in carrying charge from a to b AV= V„ - V a = ^ (10.9) Potential is measured in volts: 1 volt = 1 joule/caul Electric and Magnetic Forces 123 Considering a field due solely to a fixed charge Q, we shah compute the work done by an external agent in bringing another charge q from a great distance (infinity) in to P at a distance H from Q. As q is moved an infinitesimal distance ds along an arbitrary path (Fig. 10. .'5), it will be acted on by a practically Fig. 10.3 Calculation of potential at a point P. constant force F = qQ/4irt D r s . The work done on the system of charges by the agent exerting force F is dW = F cos 6 ds = -7-^, dr (10.10) where the negative sign comes from the fact that F and ds cos fl (= dr) are vector quantities in opposite directions. The total work done in bringing q from =o to R is then IK = ~ ( lQ f R< t= flQ 4ir«o J" r s hrtjt (10.11) Dividing by q in Eq. (10.1 1), we have the work per unit charge, which is the potential, given by v R - r„ = Q 4aW£ (10.12} Strictly, we have defined only difference of potential. Jf arbi- trarily the potential is taken to be zero at infinite separation of the charges (l'» = 0), then the potential (sometimes called the 124 Looking En: Atomic and Nuclear Physics "absolute" potential^ at a distance r from an isolated point charge Qis Q V - 4xe r (10.13) Tf q and Q are both positive (or both negative) charges, then the external agent bringing </ toward Q exerts a force in the direc- tion of the motion and does a positive amount, of work on the system. II" k(ijQ '"). This work is stored in the system as poten- tial energy qQ P = 4vt r = qV (10.14) Tin's potential energy can be recovered. If the charge q at a point distant r from Q is released, it will fly off; its potential energy is converted into kinetic energy. If q and Q have different signs, then in the trip from «; to It, the agent will have to hold q back (to prevent acceleration). The IF of Eq. (10.1 1) will be negative. Because energy is trans- ferred from the electric field to the agent, the charges are placed in a configuration of lower potential. Energy would have to be put back into the system to separate the charges again to infinity. These ideas will be used in calculating the energy stored in an atom of hydrogen, where a positive nucleus attracts the negative electron (('hap. lo). lisniiifih-. Kleetrons which leave a healed filament with negligible energy arc accelerated to pass through an aperture in a metal plate maintained at. a potential of !)00 volts above thai of the filament. What is the final s))eod. of the electrons? Each electron has a charge of — 1 .130 X 10"" coul and a mass of 9.00 X 10" 11 kg. The electron gains kinetic energy equal to the work dime on it in falling through potential difference V. (l)»w* = Ve Hence ">/?-( 2 X 1.6 X lO" 1 ' coul 901) 9.11 X 10-" kg *)'= M X I0*m/sec The kinetic energy attained by an electron in falling through a potential difference of 1 volt is given the name electron volt (ev). 1 ev - (-e)(-AV) = 1.602 X 10- ,u coul (1.00 joule/conl) = 1.002 X 10- '• joule Electric and Magnetic Forces 125 10.5 ELECTROSTATIC DEFLECTION The deflection of charged particles by electric and magnetic fields has been important in the identification of elementary particles and in the development of such useful devices as the cathode-ray oscilloscope and the mass spectrograph. As a special case, consider a parallel-plate? capacitor (Fig. 10.4) with charge —Q on the upper plate and +Q on the lower Fig. 10.4 £ = AV, s for o uniform electric field. z I + Q plate. If the distance s between the plates is small compared with the other dimensions, the electric field E is uniform in the region between the plates. If we lake a small charge </ from the upper plate across to the lower, the work done is the product of the con- stant force liq and the distance s. From the definition of potential difference, the work is also the product" of the charge q moved and AV. By equating these, A'r/s = q AV, we have E - AV for uniform field (10.15) A device for studying the charge and mass of particles con- sists of an evacuated tube in which a narrow beam of particles, defined by slits c, and <•■. passes between the plates of a parallel- plate capacitor and then impinges on a fluorescent screen S' where it produces a visible spot (Fig. 10.;">), The x component, of the velocity of a particle suffers no change as the particle passes through the capacitor and goes to the screen. In the electric field P 1 —TT AV + k-L -4— S' Fig. 10.5 Electrostatic deflection of a beam of charged particles. 126 Looking In: Atomic and Nuclear Physics of the capacitor, a positive particle will experience an upward acceleration n tf from tlie force exerted by the field E on the charge F v _ qE m m (I O.K.) The particle emerges at the right side of the capacitor with velocity components v x = v I m m v (10.17) where ( = l/v, the time required for the particle to pass through the capacitor. The particle emerges from the capacitor at an angle 8 with its original path, where . ? T.' I ten b . "JL = ■' E v x m (10.18) The deflection A C observed on the screen is the sum of the deflec- tion AB which the particle incurs while in the capacitor and deflection BC brought about by the v„ velocity component while the particle travels distance D. 2 m ir BC = Dt&n0 = ^E-,D (10.19) (10.20) Thus the measured deflection AC is as if the beam were abruptly deflected through an angle midway through the capacitor: AC = AB + BC = \ q —,+ (>E { - (d + -A tan 8 (10.21) 2 mv* mo- \ 2/ v ' If we measure AC, D, and I, we can find tan 6. Measurement of AV and $ determines E and, through Eq. (10.18), gives a value for q/mv 1 . If we know the initial speed v of the particles, we can iind a value for q/m, or vice versa. The experiment does not determine q and m separately. The experiment is usually done in such a way that is a very small angle, so that tan = (ex- pressed in radians). Then, for a given instrument, is inversely Electric and Magnetic Forces 127 proportional to the kinetic energy of the particle. Such deflection experiments are important in identifying, sorting, and utilizing charged particles. Toble 10.3 Charge-moss ratios for several particles Particle q/m, coul kg Electron Proton « particle •1.75V X I0 11 9.579 X 10 7 4.822 X 10 7 10.6 ELECTRIC CURRENT Electric charges in motion constitute an electric current. In metallic conductors there are many "free" electrons, that is, electrons not bound strongly to particular atoms of the metal. Each electron moves in an irregular path, continually colliding with atoms of the metal. If a wire is connected across a battery, an electric field is set up within the metal. The electrons tend to drift from regions of low potential to regions of high potential. This electron "wind" is the current. The continual collisions are responsible for the resistance of the metal. The kinetic energy gained by the electrons from the field and given up in collisions is l he power loss i-R which produces heating in any current- carrying conductor. In electrolytic solutions, in some types of vacuum tubes, and in certain solid-state devices, electric current may result from the Positive plate +. — Negative plate |t ,/) Ammeter Electron flow High potential Conventlal current Low potential Fig. 10.6 Direction of electron flow and of con- ventional current. 128 Looking In: Atomic and Nuclear Physics motion of both positive and negative charges. Any currenl direc- tion is a convention; the choice of sign is arbitrary. In this honk we shall regard the direction of conventional current as that of the flow of positive electricity. The conventional current is from high potential to low potential ("from + to — "} in the external circuit. If two parallel wires carrying current in the same direction are brought near each other, they attract each other. This effect is the basis for the definition of the ampere, the inks unit of current (Sec. 10.2). This attraction is not an electrostatic (coulomb) force between unbalanced charges. It is a magnetic force arising from the motion of charges. It is convenient to discuss these magnetic forces in terms of a field. 10.7 MAGNETIC INDUCTION The basic vector for describing a magnetic field is called the mag- netic induction, B. (Magnetic field strength would be an appro- priate name for B, but historically this name has been assigned to another vector H connected with magnetic fields.) We identi- fied and measured an electrostatic field in terms of the force exerted on a unit positive charge. Kxperimetitally we identify the presence of a magnetic field from the fact that if a magnetic field is present, a moving electric charge will experience a sideways magnetic force. The magnetic induction H is defined as the vector which satis- fies the relation F = q(v X It) (10.22) where force F, charge q, and velocity v are the measured quanti- ties. This notation of a "vector cross product" means that F is Fig. t0.7 F = q(v X B) Electric and Magnetic Forces 129 perpendicular to both v and B and directed so that, if the fingers of the right hand are directed from the direction of v (around through an angle of less than 180°) to that of B, the right thumb will point in the direction of F. The magnitude of F is given by F^qvIixmB (10.23) where 8 is the angle included between the positive directions of v and B, Xote that a vector cross product v x B is zero if v is parallel or antiparallcl to B, Notice also that A x B = — B x A; that is, A X B is equal in magnitude but opposite in direction to B X A. The unit for B, from Eq. (10.22) is newton/(coulomb meter/ second). This inks unit for B is given the name weber/meter 1 we her _ newton m 2 eoul m/sec - 1 newton amp-m An earlier cgs unit for H, still often used, is the gauss. weber 1 nv = 10* gauss The summation of B over a surface J'B ■ r/s is called the magnetic flux *. The weber is the unit of flux. When a charged particle moves through a region in which both electric and magnetic fields are present, the resultant force on the particle is given by F = ryE + f/(v X B) (10.24) Only in the special case where E, v, and B are suitably oriented can we replace Eq. (10.24) by a scalar equation which suggests straight, addition: F = q E + qv B sin B (10.25) where is the angle included between the positive directions of v and B. 10.8 MAGNETIC FORCE ON A CURRENT An electric current may be visualized as moving charges. Assume that in a conductor of length I there are n conduction electrons per unit volume, each with charge q and each having an average drift speed i\ (The negative electrons drift in a direction opposite 130 Looking In: Atomic and Nuclear Physics to that of the current, Fig. 10.8.) The distance an electron moves per second is v. The volume of charge passing a certain cross sec- tion A of the wire is Av. In this volume there are nAv = JV con- duction electrons. If this conductor is in a uniform magnetic / U 5 Fig. 10.8 Representation of an electric current. induction B, the force q(v xB) on each moving charge in the conductor produces a force on the conductor of length / which is F = Nq(v X B) But the velocity is 1//, and Nq/t is the current i, so the equation for the force hecomes F = Nq (j X b) = ( N f\ (1 X B) = »(| X B) (10.26) or F = Bil sin d 10.9 MAGNETIC DEFLECTION OF CHARGED PARTICLES The force on a charged particle moving in a magnetic induction is at right angles to B and to v. The particle is accelerated, I nit always perpendicular to its velocity. The magnetic force changes the velocity (vector) but not the speed (scalar). No work is done on the particle by the magnetic force, for cos 9 = in the expres- sion W = /F cos da. When a charged particle enters a uniform held with its ve- locity perpendicular to B, the particle experiences an acceleration of constant magnitude qvB/m perpendicular to its velocity. The particle describes a circular path with constant speed. The cen- Fig. 10.9 Path of charged particle in plane normal to 1i. Electric and Magnetic Forces 131 tripetal force needed to keep the particle in circular motion is supplied by the magnetic side thrust. Since the velocity of the particle is always perpendicular to the induction, sin 8 = 1 in Eq. (10.25). Newton's law F = ma then can be expressed as The momentum of the particle mv can be found if we know B and q and measure r. If measurements of electrostatic deflection and magnetic deflection are carried out on the same beam of charged particles, one can determine both q/m and v for the particles. In this way Thomson measured the charge/ mass ratio for electrons in 1897 (Chap. 11). Similar deflection methods arc used today in some types of mass spectrometers to obtain accu- rate values of q/m for ions and isotopes. 10.10 MAGNETIC INDUCTION OF A CURRENT We have just considered problems relating to the forces exerted by a magnetic induction on a moving charge or on a current- carrying conductor. A second class of problems involving mag- netic lields concerns the production of a magnetic induction by a current-carrying conductor or by moving charges. The relationship between current i and magnetic induction B is given by Ampere's law. The magnetic induction at a point P arising from a current i in a wire is the vector sum of contribu- tions from every element of the wire. The induction at P due to the current in element d\ of the wire is d\\ - I \ -_ til X r Air r' (10.27) We have again used the notation of the vector cross product. The magnitude of rfB is given by dB - idl 4w r 2 u., sin 8 (10.28) The constant u,. is called the permeability of free space. In the mks system its value is /in = 4jr X 10 -7 joule/amp*-m 132 Looking In : Atomic and Nuclear Physics The direction of r/B is perpendicular to the plane of the vectors dl and r, and such that if the fingers of the right hand arc turned (through an angle less than 180°) from dl to r, then the right thumb will point in the direction of dli. Fig. 10.10 Induction dU contributed by cur- rent element i dl. Ampere's law as expressed in the last two equations cannot be subject to direct experimental check, for we cannot isolate an element i dl of an electric circuit. Actually Ampere's law was not deduced from any single experiment. Rather it summarizes many experiments dealing with magnetic effects of circuits of different geometry and witli magnetic forces exerted by currents on each other. To illustrate the tise of Ampere's law we shall calculate the magnetic induction (I) at the center of a circular loop, and (2) at a point near a long straight conductor. At the center of the loop (Fig. 10.11) the direction of the magnetic induction H is Fig. 10.11 Induction B (out of page) at center of a circular loop. Electric and Magnetic Forces 133 perpendicular to the plane of the current elements i dl and r, in the sense given by the cross product d\ x r, or out of the page. Since a radius to any point on the loop is perpendicular to the tangent to the circle at that point, sin = 1 in Bq. (10.28). Writing r d<p for dl, the magnitude of B at the center of the loop is B = toL-* r*"= ■>,■ 0&»> To calculate the magnetic induction at a distance R from an infinitely long straight wire (Fig. 10.12), it is convenient to take Fig. 10.12 Induction B near a Mroight conductor. the z axis along (he wire in the direction of / and with the origin O at the point of the wire closest to l\ The upper and lower halves of the wire make equal contributions so we may compute B by integration of the expression B = 2 f- u i /-*. sin '/: 4x } - » r For the section of wire below the origin (z < 0) we have (10.30) -R — tan 8 and R — sin 6 dz = d(.^\=«4 \tm 0} sin 2 de I Equation (10.30) may be written in terms of one variable 0: R _ tint [*ft . . sin 2 e R d0 noi r*te . 134 Looking In: Atomic and Nuclear Physics giving B = li.nl 2-xii (10.32'] 10,11 DIRECTION RULE The relative directions of the vector quantities in Eqs. (10.22) and (10.27) implicit in the vector cross products can be remem- bered conveniently from the following rules: (a) If, in imagination, the right hand grasps a current- =^z^ (a) (M Fig. 10.13 Magnetic induction B near an electric current: {a} Side view, i toward right; (b) end view, j out. It counterclockwise. Current "in" Current "out" B B, © (a) (b) (c) Fig. 10,14 Force on a current-carrying conductor is from strong field region toward weak field region. carrying conductor with the thumb pointing in the direction of the conventional (+) current, the fingers encircle the wire in the same sense as the magnetic induction B. Thus in rig. 10.13a, B is out of the page above the wire (indicated bj dots) ; B is into the page below the wire (indicated by crosses). Electric and Magnetic Forces 135 (h) When a current-carrying wire is in a magnetic field, the magnetic force on the wire is directed from the region of stronger induction toward the region of weaker induction. Consider a wire perpendicular to the page carrying a current into the page (Fig. 10.14a). The local magnetic induction encir- cles the current in the clockwise sense. If now this current-carry- ing conductor is placed in an external field B, (Fig. 10. 1 4b), B and B\ reinforce each other above the wire (strong field region) and partly cancel each other below the wire (weak field region). The force F on the wire is down. A representation of the net field due to B and Bj is shown in Fig. 10.14c, 10.12 INDUCED CURRENTS A further important relation between magnetic fields and electric current is the principle of induced emfs on which the design of generators, transformers, and motors is based. Michael Faraday and Joseph Henry, at about the same time (1831), showed that an emf is induced in a conductor when there is any change of mag- netic flux linked by the conductor. It is convenient to consider this single principle from two viewpoints, (i) An emf e is induced whenever a conductor moves across a magnetic field e = IvB sin d (10.33) where I = length of wire v = its velocity — angle between » and B When mks units are used on the right of Eq. (10.33), e is given in joules per coulomb or in volts. (2) An emf is induced whenever the flux ($ = BA) changes through a circuit: e = — d* dt (10.34) I f the rate of change of flux is in webers per second, the emf e is in volts. Lena's law states that whenever an emf is induced, the induced current is in such a direction as to oppose (by its magnetic effects) 136 Looking In: Atomic and Nuclear Physics the change inducing the current. Lena's law is really an example of the conservation of energy principle. As applied to Fig. 10. l. r >, Lea* 's law says that the direction of current induced in the moving wire must he Mich as to oppose its motion. This requires the magnetic force on the wire to be toward Fig. t0,!5 Emf induced in wire ob moving across a uniform magnetic field. the left. From Sec. 10.1 1 b, the net induction ahead of the wire (at right) must be greater than that behind the wire (at left). To reinforce the external li, directed out, ahead of the wire, the induced current must be down, from a to b, from tint right-hand rule (Sec. 10.1 In). a II b Fig. 10,16 Emf induced in a coil moving from a to b in nonuniform magnetic field. Note that in this "generator" b is at a higher potential than a. Positive charge is forced to flow from a (low potential) to b (high potential) by the work an external agent does in moving the wire ab against the magnetic force. In the external circuit bca, the charge flows from high potential toward low potential ("from + to — "); it can do useful work, and it produces ,-/>' heating in the conductor. Electric and Magnetic Forces 137 QUESTIONS AND PROBLEMS 1. State some similarities and some differences between the phe- nomena of electric fields and gravitational fields. 2. A positively charged rod is brought near a ball suspended by a silk thread. The hall is attracted by the rod. Does this indicate that the ball has a negative charge? Justify your answer. Would an observed force of repulsion be a more conclusive proof of the nature of the charge on the ball? Why? 3. The circuit in the diagram consists of two concentric circular sec- tions AC and l)E and two radial sections CI) and HA. There is a cur- Fig. 10.17 rent i in the direction shown. Starting from Kq, (I0.2S). derive an expression for the flux density at O, the common center of the arcs. Show the limits of integration arid any special values of factors in die equation. Show clearly the contribution of each part of the circuit. What is the direction of the (lux density at 0? Ans. ? ol(fii ~ _ ^ij 4. A 5.0-m straight wire ab (Fig. 10.18} is allowed to fall through a uniform magnetic induction of 2.0 XII) 5 webcr/m* directed pcrpen- Fig. 10.18 x a V dicular to the wire, (n) What is the emf induced in the wire at the instant its speed is 3.6 m/see? (6) What is lire direction of this emf in the wire? (c) Which end of the wire is at the higher potential? Am. 3.6 X I0-* volts; toward right: 1',. > l'„ 5. A Ferris wheel is 100 ft in diameter. Ms axis of rotation i- on a north-south line, (a) If the horizontal component of the earth's field is 2.00 X 10 _i weber/m* and the wheel is rotating at 2.00 rev/rain, what i* the potential difference existing between the axle and the end of one apoke? (b) Which i^ at the higher potential? Ans. 488 pv (}. Assume thai this mom has a uniform magnetic field directed vertically downward and of flux density B = 1.0 X 10-" weber/m*. (a) Determine the magnitude and direction of the force on an electron 138 Looking In: Atomic and Nuclear Physics thai enters the room moving due east and at an angle of 30° above the horizontal with a speed of 3.0 X 10* m/sec. What, will be the speed of this electron 1.0 X 10 '■' see after if enters the room? (b) An east-west wire is stretched horizontally across the room. What will be the direction and magnitude of the force on :i (i.O-m section oT the wire when there is a current, of 12 amp westward in the wire? Am. 2.4 X 10-" newton, south; 3.0 X 10* m/sec; 7.2 X 10" 1 newton, sou til The most important thing a young man can learn from his first course in physics is an appreciation of the need for precise ideas. W. S. Franklin A new principle is an inexhaustible source of new views. Marquis de Vauvenargues To succeed in science it is necessary to receive the tradition of those who have gone before us. In science, more perhaps than in any other study, the dead and the living are one. Charles Singer 11 The Electron The electron has conquered Physics, and many worship the new idol rather blindly, H. Poincare It is the purpose of this chapter to discuss the experimental evi- dence of the existence of the electron, some of the measurements which have been made on it, and the limitations of the classical free-electron theory of the conduction of electricity in metals. 11.1 IDENTIFICATION OF THE ELECTRON In a paper "On Cathode Rays" (1897), J. J. Thomson first estab- lished the existence of free electrons. Thomson's investigation was prompted by ihe divergent opinions people had at that time about the nature of cathode rays. Experimenters had shown that at low pressures (about 0.01 mm of mercury) air becomes a good conductor of electricity and that, the discharge of electricity through a gas produces light whose color depends on the gas and in a pattern which depends on voltage and gas pressure (Kig. 11.1). Some people considered the rays charged particles; others thought of the display as a phenomenon in the "aether." Thomson 139 140 Looking In: Atomic and Nuclear Physics suggested that an explanation based on particles as :i working hypothesis was more likely to he successful and could he more easily tested hy known laws (of mechanics) than any explanation based on properties of the aether about which little was known. Mr therefore devised experiments "to test some of the conse- Crooke's dork space Cathode / _ Faraday dork spoce 7 #- !■ % Positive column -A Anode Negative glow II Anode glow To pump Cathode glow ' Fig, 11.1 Discharge of electricity through a gas at reduced pressure. quences of the electrified particle theory." The objects of Thom- son's experiments were: 1. To verify that cathode rays carry a charge, a charge which accompanies the rays when they are deflected by a magnetic field 2. To investigate quantitatively the deflection of cathode rays in an electric field, which deflection also indicates the presence oT charge :?. To determine the energy of the cathode rays and, by using this value with data on the magnetic deflection, to deter- mine the speed and the ratio of charge to mass e/tn, for the "particles" 4. To determine speed and <■ hi also from a combination of elec- tric and magnetic deflections Thomson obtained information on each of these properties in the following ways: 1. In a tube such as shown in Fig. 11.2, cathode rays leave the cathode C, pass through an opening in the anode A, and reach a region B where they are deflected by a magnetic field, pass through an opening in a grounded cylinder 6, and finally reach a collecting conductor /' mounted inside that cylinder. An increase in charge is registered by an electrometer connected to /' only when rays enter the opening in Q. The observations prove that a charge is inseparably connected with the cathode rays and that it is a negative charge. The Electron Mi 2. Deflection by an electric field was investigated in the tube shown in Fig. 11.3, the precursor of our modern cathode-ray oscilloscope. It verified the negative charge of cathode rays. 3. The energy of the cathode rays was measured in a tube without deflection plates and provided with a screened electrode, Fig. 11,2 Discharge tube far demon- strating the (negative! charge and magnetic deflection of cathode rays. The magnetic field is excited in space S by coils placed outside the tube. as in Fig. 11.2. The innermost, electrode contained a thermo- couple. Its increase in temperature in a given time was measured, and simultaneous measurement was made of the charge from the rays received at the thermocouple. If, in the time considered, A r particles strike the thermocouple each bearing a charge r, the total charge is Q = A>. From the ri.-e in temperature the total energy is known Ek - S\mt:~ (i i.|) Fig. 11.3 Thomson's tube for electricol deflection of cathode rays. 142 Looking In: Atomic and Nuclear Physics where m is the mass and v the speed of one particle. Next the radius r is measured for the path described by the particle in a magnetic field. The centripetal force is equated to the magnetic thrust ™ = cvB T (11.2) For each of the two flat circular coils used to produce the mag- netic field, B is given by Eq. (10.29) o _ Mo/ B - "27 (11.8) where J is the current and mu is the permeability of free space, 4w 10" 7 weber/amp-m. It follows from the preceding equations thai me* Nmv 1 2E k v — and erli NerB QnJ 2E k m tB Qho2P (11.4) (11.5) With different gases (air, II 2 , and CO.) in the tube, Thomson showed that e/m had the same value 2.2 X 10" coul/kg. Thus cathode-ray particles are independent of the nature of the gas. 4. Values of e/m can be obtained by a different method for comparison with the foregoing results. In the tube of Fig. 11.3, a magnetic field is established, into the plane of the paper, by two coils whose diameters are etjual to the length of the capacitor plates. The crossed electric and magnetic fields are adjusted to give the cathode-ray zero deflection. The cancellation of the electric force (upward) and the magnetic force (downward) is expressed by isV — Bicv (11.0) where A'i is the electric field intensity. Next the particles are deflected by a magnetic field only, directed perpendicular to their velocity, and the radius r of the path is determined from observed deflection on the screen. Here — = Btfo r (11.7) From Eqs. (11.0) and (1 1.7) e v Ei m v ,B, rBJi-, The Electron 143 (11.8) With this method, and using air, C0 2 and H 2 , Thomson obtained similar values for e/m. Again, the nature of the gas did not influence e/m. 11.2 THE CATHODE -RAY TUBE We digress to point out the relationship between Thomson's apparatus and the modern cathode-ray tube. If the potential to be observed is applied to plates !\ (Fig. 11.4) and a potential -o p. (a) Fig. 11.4 (o) Cathode ray tube, (b) Test and sweep potentials. that increases linearly with time is applied to the horizontally deflecting plates /■*., the electron beam (cathode ray) traces out a wave form. If a saw-tooth wave form is repeated at the same frequency as the alternating potential on P t , the trace oi\ the screen is repeated each cycle and appeal's to be stationary. 144 Looking In: Atomic and Nuclear Physics 11.3 ELECTROLYSIS AND THE ELECTRON It has l>een seen (Chap. 8) that the combining properties of the elements can l>o interpreted in terms of an atomic theory of matter. Faraday's study of the electrolysis of aqueous solutions of chemical compounds suggested thai electricity is also "atomic" in nature. Faraday's discoveries of fundamental importance may be expressed thus: 1. The mass of a substance separated in electrolysis is propor- tional to the quantity Q of the electricity that passes. 2. The mass ;!/ of a substance deposited is proportional to the chemical equivalent k of the ion, that is, to the atomic mass A of the ion divided by its valence t>, M-kQ-^Q (11.9) where /"' is a constant of proportionality known as Faraday's constant. Careful measurements have been made of the amount of electricity required to lilierate a mass of any substance numeri- cally equal to its chemical equivalent, say 107.88 kg of silver, 1 .008 kg of hydrogen, or 05.38/2 kg of ainc. This value is 9.052 X 10 7 coul/kmole. It is called 1 faraday. It is represented by F in Eq. (11.9). Faraday inferred from his experiments that the same definite amount of electricity is associated in the process of electrolysis with one atom of each of these substances. He considered that this charge is carried by the atom, or in some eases by a group of atoms, and he called the atom or group of atoms with its charge an ion. In 1874, Stoney stated the hypothesis that "nature pre- sents us with a single definite quantity of electricity." He sug- gest ed the name electron for this quantity and calculated its value from the faraday and from Avogadro's number N*. In terms of values now accepted, /.' = jy,«e . ij,cjf)2 x 10' coul/kmole 9.052 X 10' coul /kmole _ -„ . e ~ 0.0219 X m molecules/kmoie , ' 6021 X l ° C ° Ui The Electron 145 11.4 CHARGE OF AN ELECTRON, MtLLIKAN'S EXPERIMENT Millikan, about 1909, devised a highly precise experiment based on the fact that electrically charged droplets of oil can be held stationary between the horizontal plates of a capacitor by adjust- ing the voltage between the plates so that the weight of the drop is balanced by the force due to the electric field. The "oil drop" experiment can be used (1) to show that electric charge occurs in multiples of a discrete amount and (2) to measure the value of the smallest charge, the electron. See Fig. 1 1.5, page 147. An oil drop will fall with accelerated motion until the drag due to air viscosity becomes great enough to balance the weight of the drop. For the small speeds which occur in this experiment, the frictioual drag is proportional to the speed of the drop. Setting the weight mg equal to the f fictional drag fct>) gives us an expres- sion for the terminal speed vi reached by the drop falling mg = h'i (11.10) To evaluate k, we take from hydrodynamics Stoke'slaw/.' = t'wnjr, where ij is the viscosity of air and r is the radius of the drop, assumed spherical. If the oil drop has a charge c/, due to an excess (or deficiency) of electrons, and if a uniform electric field is now established between the capacitor plates, the electric force on the drop is F-ft-J, (li.ll) When the drop now falls under the influence of gravitational and electric fields, it attains a new terminal velocity v^: F — mg = ft»j O'o up taken as positive) (11.12) Combining Kqs. (11.11) and (11.12) gives V --,<} — mg = hit (11.13) Suppose now that, owing to random ionization, the charge on the drop increases by an amount q„. There will be no significant change in mass, but a new terminal speed v 3 will result from the change in electrical force: y -j (<1 + 9») — mg = Icv-s (11.14) 146 Looking In: Atomic and Nuclear Physics From Eqs. (11.13) and (11.14), we have a measure of the change in charge in terms of observed speeds: Qn = y &(->» - *>j) (11.15) Millikan observed that experimental values for q„ were always whole-number multiples of a certain quantity. Pie inferred that this quantity is the basic unit of charge, the electron. Example. By timing its full through a known distance, an experimenter determines "the successive -prods y„ and c„ +1 of a single oil drop having successive different random charges. He computes the change in charge (q„) from Kq. (11.15) as tabulated below. What do these data indicate for the charge of an electron? By inspection, one notes that the values for </„ are, within experi- mental uncertainty, whole-number multiples of 1.6 X 10"". By divid- ing this number into the charges q„, we find the values n in the second column. The experimental value for the electronic charge is (hen the average of the values of e in the last column. q,„ X10-" coul 1.6 X 10~" n qn e = — » n X 10 "coul 4.76 2.98 3 1.59 3.21 2.00 2 1.61 4.96 3.10 3 1.65 8.07 5.05 5 e 1.61 = 1.61 11.5 ELECTRONS IN METALS The conduction of electricity through metals is fundamentally different from electrolytic conduction. When a copper wire has carried an electric current, even for a long time, no chemical change can be detected in the copper. .More than a century ago it was first assumed that electricity is an agent that can flow freely in a metal. The sign of this charge and its direction of How were unknown, but the flow was assumed to be from the arbi- trarily defined positive (high-potential) terminal to the negative (low-potential) terminal. At present we have evidence that elec- trons flowing in the opposite direction are responsible for the con- duction of electricity in metals. These electrons are called free electrons because they are temporarily detached from atoms. The number and freedom of motion of these electrons determine The Electron 147 the properties of the material as a conductor of electricity (and heat). Xo other property of solids has such an enormous range of values as does electrical conductivity (Fig. 11.6). The best con- ductors have electrical conductivities greater than that of the best insulators by a factor of 10". ^=feo v Telescope Fig. 11.5 Millikan's apparatus for determining the charge of an electron. Conduction electrons have been compared to a gas which is free to move within the metal, under the influence of an applied electric field. The metal is visualized as consisting of an assembly of stationary positive ions permeated by an electron gas which makes the metal as a whole neutral. This qualitatively attractive picture proves to be inadequate in several important respects. first, on this basis we should expect the specific heat of a metal to consist of two parts: that of the ionized atoms considered as vibrators (3/2) and that of the electron gas (}!£), or a total specific heat of f ft. This finding is in marked disagreement with 10 _ 10" 10° io J 10 6 10' to" io ,: Ohm-cm 10" 10* =f Cu Ni Ho Si Ge n i i i r Se CU2O 8 Celluloid Mica ZnO Glass Amber Paraffin Porcelain Quortz Ceraml -Semiconductors - ■ Insulators ■ ■-*- Conductors™ Fig, 11,6 Resistivities (ohm-centimeters) vary over the enormous range of aboyt lO. 2 " Looking In: Atomic and Nuclear Physics 148 measured specific heat, of dements in the solid state, namely, 5S cal . P ( mole degree) = 3/f . a relation discern! by Du hmg and Petit as early as 1810. Second, the variations of e tec tncal conductivity with temperature and the escape of electrons through a metal surface at high temperature (thermion.c emis- sion) cannot be explained quantitatively on a classical e c ct on gas theory, neither can the great variation m conductivity. Quantum theory (Chap. 16) provides a satisfactory way out of these difficulties. QUESTIONS AND PROBLEMS 1 (a) What Ls the kinetic enemy of at, electron which moves in a oirce" of radius 5.2 em perpendicular to a magnetic mi uction of 9 .1 TlO^weWom*? (6) 523 potential difie^nce would be reqmred to .top this electron after i, ^fctff'fi^ ,j X l0 ' volts 2 Vn electron of mas « and charge B falls through a potential difference 1' and .hen enters at right angles, a region of uniform mag- fiS*. 8. What is the radius of the electron pa* m^mag- " L,t 3 Etetrom traveling 2.00 X 10' m/see are subjected to a magnetic i,uha-tb f 0.0030 weber/m» in the apparatus „^ by Thomson £K3> to Snee/Ma) The capacitor plate '■^««J"J- wSt voltage imiM be applied to them to return the beam to its m- J ^teJ 1 StioBl (») SL in a .ketch the ^*£~«*» eleclrnu velocity r. the maguel.c induction ft and the Uu tnUuld fc.^ 4. An oil droplet of mass 2.5 X 10"" gm. which <^ "MH^d 2 electron charges, is between two horizontal capacitor p ate* 2 em JpSZnS ft* *e droptef is entirely.snpported by etectne foraefe from falling, what must be the potential ddTerenrc holi v^^Pjg 10 rft *SKb- photograph, an ejeotron pato* bjrfWo a circle of 12-cm radius bv a magnetic induction of O.OOsO weber/m^ ?«i<W»lutP the enemy of the electron (*) Calcdate tho enemy of an Hec.ron whose path radius is 20 cm in the *.,,,, {«• d- /Ins. 3.6 X 10-' 6 joule or 2.2 Mev, ^.U X iw J"""- 6 V narrow beam of electrons traveling with speed e along toe x & t pa s^s etwecn the horizontal plates of a parallel-plate eapacrig F ,3 Htamed lo a potential difference V. The separation of the plan-s 111 Show that in traveling a distance ( in the capuclor the beam will experience a deflection from J lie x axis, given by „ t V I 9 = - T ~\ 12 Ions and Isotopes The most important trends in indus- try today spring from an increasing knowledge of the properties of atoms and their component parts. David Sarnoff The discovery of isotopes was foreshadowed by studies in radio- activity (Chap. 17) about 1007. The possibility that two different radioelements might lie identical chemically was inferred by the failure to separate certain ones by any chemical means available. Also Thomson's study of positive rays (HJl.Tj in ion-deflection experiments yielded two lines for neon (atomic masses 20 and 22) : yet no dilTereuees were observed in the optical spectrum of the Ne gas. Thomson recorded his suspicion that "the two gases, all hough of different atomic weights, may be indistinguishable in their chemical and .spectroscopic properties." 12.1 MASS SPECTROMETERS Boon after World War I, Dempster, Aston, Bainhridge, and others devised instruments for determining both the masses of isotopes 149 150 Looking In: Atomic and Nuclear Physics and their relative abundance. Although there are many types of mass spectrometers,* a brief consideration of their common ele- ments should clarify the operation of any type. A moving particle might be characterized by its velocity v, its momentum mv, or its kinetic energy \rns-. We may consider arrangements of electric and magnetic lie Ids designed to sort charged particles according to these properties. Two types of energy selector are suggested in Fig. 12.1. In the first, ions from a source 8 are accelerated through a potential i.ri J""\ {a) Fig. 12.1 Energy selectors. (6) — :— difference V. They acquire kinetic energy ^mu 2 equal to Vq. The speed of an emerging ion is then \ m (12.1) We have tacitly assumed that the ions are at rest at s. Actually they may have small (I ev) energies of thermal motion. This is usually negligible compared with the energy {say, >100cv) ob- 'ained from the electric field. But for some purposes, the slight spread in the velocity values for emerging particles might have, to lx> considered. In Fig. 12.16, the ion beam passes between the plates of a curved capacitor. The ions are acted upon by an elec- tric field /■-' in the direction of O. The ions move in a circular arc. The centripetal force is provided by the electric field: mv* r. - W = Eq (12.2) * in a mass spectrometer, a meter measures an ion current; in a mass Spectrograph, the record is obtained on a photographic plate. Ions and Isotopes 151 This equation can be rearranged to show that the device is an energy selector &nv* = %REq (12.3) In the selector of Fig. 1 2.2, the beam of positive ions passes through a region where a magnetic induction B is directed out- Fig. 12.2 Momentum selector (B is directed out; /V'.lJ E = 0). ,' -A. R r < * v « bid N o ward from the page. By equating the centripetal force to the mag- netic side thrust, we get mi- li = qvB (12.4) and it is apparent that this tie vice is a momentum selector mv = RBq (12,5) Consider next a twain of positive ions acted on by an electric field li and a magnetic induction B at right angles to each other. In the situation of Fig. 12.3, the ions experience an upward force B„„, Fig. 12.3 Velocity selector. 1 + — *■ I fr. I qE due to the electric field. With B directed out from the page, 1 he- magnetic side thrust on the ions is qvB, downward. Only those ions for which qii = qvB (I2.fi; will pass through without any deviation. This filter selects ions of a particular velocity E S = B (12.7) 152 Looking In: Atomic and Nuclear Physics I» a Baiubridge mass .spectrograph (Fig. 12.4), positive ions from an ion source arc colli mated by slits s, and x* and then pass into a region in which they experience an electric force to the left and a magnetic force to the right (supplied by Hi which is directed into the page). From Etj. (12.7), the only ions which pass through slit « 3 are those of speed r = - Bi (12.8) where /; has Iktu expressed as V ".-,-. Beyond s ;i the ion is influenced only by a uniform induction B» (into the page) which causes the Fig. 12.4 Boinbridge moss speetro- groph. ion to move in a circular arc of radius ft until it strikes the photo- graphic plate /' where it makes a developable trace. By measuring the distance 2/1 from this trace to slit s a , and using Eq. (12.4), we can find the charge mass ratio '/ _„ '' m BJl V B y B#R (12.9) The charge on the ion will he a multiple of the charge on the, electron (usually U or 2c), which the experimenter must find. With q known, he can calculate the mass m of the ion. lie may then add the mass of the missing elect ron(s) to (hid the mass of the atom. For measurements of highest precision (a few parts in I0 7 }, a mass spectrometer is designed to cover only a limited region in Ions and Isotopes 153 the mass scale from I to 2">0. Unknown masses are found by inter- polation between known masses (often using ions of molecules as well as of atoms). Also, atomic masses can be deduced from the energy release in certain kinds of nuclear reactions (Chap. 18). A table of the "Ix-st" values of atomic masses obtained as aver- ages of mass spectrometer and nuclear reaction data, adjusted for self-consistency, is useful in many calculations in modern physics. * 12.2 ISOTOPES As the accuracy in measurement of atomic masses increased, it was established that not. all atoms of the same element have the same mass. Atoms of the same element (same Z) which have dif- ferent masses are called isotopes. Many elements (Be, F, Xa, Al, P, Co, etc.) occur naturally with only one isotope. Many others (H, He, Li, B, etc.) have two, and tin, the most varied, has no less than 10 isotopes. It the mass of a carbon atom is taken to be exactly 12, then the masses of the other elements, determined by quantitative chemical analysis, come out to be nearly whole numbers. Histori- cally this led to Front's hypothesis that all elements were built from hydrogen. This picture was spoiled by certain atomic masses determined chemically: 35.8 for CI, 63.54 for Cu, etc. But when measurement of isotope masses became possible, it was found that the mass of every isotope of every element was very close in an integer on the scale in which carbon is taken as 12 antra (or, originally, oxygen defined as 10 amu). Naturally occurring chlo- rine, for example, is a mixture of about 7o per cent of an isotope :S4.!)7!K) and 25 per cent of an isotope 38.9773. Its average mass, as found in chemical experiments, is then §(35) + t($~) = 35.5. The whole-number rule may be retained in this form: The mass of every isotope of every element is well within 1 per cent of a whole number when expressed in atomic mass units, defined by taking carbon as 12 amu, exactly. We thus retain the picture of all atoms built up of some unit of which there is 1 in hydrogen, 4 in helium, Hi in oxygen, etc. We have yet to explain, however, * Sec (.'. II, Blunt-hard, C. R. Burnett, H. G. Stoncr. and R. L Weber, "Introduction to Modern Physios," appendix fi, pp. 392 100, l'rciitire- Hall, liic, Engleivood Cliffs, X. J. , 1958. 154 Looking In: Atomic and Nuclear Physics why the atomic masses arc not exactly integers instead of being very nearly integers. QUESTIONS AND PROBLEMS 1. The values of E and B in the velocity selector of a mass spec- trometer are MIDI) colts, n. and 0,050 weber/m*. (fl) Wluit will be the speed of ions passing through this selector? (b) By what radius will a singly riiarsnl ion of mass 50 amu be deflected by a magnetic field of 2.5 X 10~ a weber/m 1 after leaving the velocity selector? Arts. 1.6 X 10' m/aee;2.1 em 2. A singly charged positive ion is accelerated through a potential difference of 1,000 volts, li is then subjected to a magnetic field of 0.10 weber/m 1 in which it is deflected into a circular path of radius 18.2 cm. (a) What is the mass of the ion? (h) What is the mass number of the ion? Am. 2.05 X 10 ■'■ kg. 15.9S amu: If. 3. A dust particle has a mass of 3.0 X 10~ s kg and a charge of 5.0 X 10 10 coul. The particle is accelerated in an electric held until it has a speed of 4.0 m/sec. (a) Calculate its kinetic energy in joules, (b) What potential difference is required to give the particle this speed? (c) If the particle moves at right angles to a magnetic induction of 0.20 weber/m 2 , what force will the particle experience? (</) What is the radius of the circular path in which the particle will move in this mag- netic field? Ans. 2.4 X 10"' joule; 480 volts; 4.0 X lO" 10 newtnn; 1.2 X 10 a m 4. In a Dempster mass spectrometer, positive ions formed by heating a salt of an element are accelerated to a slit s, by a potential difference V (about 1,000 vol Is). A narrow bundle of ions then passes Photo, plote or electronic detector Fig. 12.5 through the slit into a semicircular chamber where there is a magnetic induction II perpendicular to the ion velocity i> (Fig. 12.5). Ions having different values of e,'m will travel arcs or different radius. Show that the charge/mass quotient of an ion can be computed from e m 2F fi ! r 2 Ions and Isotopes 155 5. On the photographic plate of a ma>s spectrograph, a trace made by a singly charged ion is h.und jusf halfway between the line formed by 16 (+) and that formed by CH,(+). Find (he mass of this ion if the mass sped rennet er is (u) a Bainbridge type, when 1 the mass of an ion is proportional In the radius m = kr, and (b) a Dempster type, where the dispersion equation is »i = kr 1 . Ans. 10.018148 amu; 1B.009074 amu 6. In a method devised by S. A. Goudsmit, masses of heavy ions are determined by timing (heir period of circulation in a known mag- netic field. To get an idea of the timing requirements, calculate the period of revolution of a singly charged ion of iodine ail 127 (mass 126.945 amu) in an induction of 0.045 weber/m 5 . -4ns, about 1.8 X I0~* sec 7. Show why the mass spectograph gives data on the atomic masses of individual ions, while conventional chemical methods yield results only on average atomic, masses. 8. Silicon has an atomic number of II. Consider two isotopes of silicon having mass numbers 28 and 30, fill in ihe remaining spaces in the table: Mass number. , 28 30 Number of electrons in the atom Number of positive charges in the nucleus Number of protons in the nucleus Number of neutrons in the nucleus. 13 Electromagnetic Radiation Electricity, carrier of light and power, devourer of time and space, bearer of human speech over land and sea, greatest servant of man. Charles Eliot I. CLASSICAL THEORY In 1 8(14 , James Clerk Maxwell completed the structure of classi- cal electric and magnetic theory. His summarizing equations stand with Newton's laws of motion and the- laws of thermody- namics as masterpieces of intellectual achievement. The four dif- ferential equations show how electric and magnetic fields are related to the charges and currents present and how - they are related to each other. They correlated experiments in large areas of physics and predicted important new results. Specifically, Maxwell showed that a changing current will radiate electromag- netic waves in which E and li are perpendicular to each other and to the direction of the wave motion. Tlis theory predicted that electromagnetic waves of all frequencies should travel with tiie speed of light c, whose numerical value can be determined by 156 Electromagnetic Radiation 157 measuring the force between currents. This theory was experi- mentally verified by Hertz in 1888, and by 1901 Marconi suc- ceeded in transmitting electromagnet ic signals across the Atlantic Ocean. The. electric generator, motor, betatron, television, and radar are based on principles included in Maxwell's equations. Home of the relations and experimental facts which Maxwell synthesized carry the names of earlier investigators,* 13.T GAUSS' LAW FOR AN ELECTRIC FIELD Imagine a potatolike surface immersed in an electric field. The flux *« of the electric field through this arbitrary surface is measured by the number of lines of electric force that cut through Fig. 13.1 surface. Electric flux through o the surface. Let the surface be divided into elementary squares An small enough so they may be considered to be plane. An cle- ment of area can be represented by a vector As whose magnitude is proportional to the area: the direction of .is is taken as the outwarddrawn normal to the surface element. The field intensity E is practically constant over an element- As; B is the angle between K and As. The flux is found by adding up the scalar product A* ■ As cos 8 for all the elements into which the surface has been divided: *k = 2E-As (13.1) For a more precise definition we replace the sum by an integral: * K - /E • r/s (13.2) The integration is to be taken over a closed surface. * A reader not familiar with calculus notation may wish (o skhn Sees. 13.1 to 13.4 and resume his study immediately after the statement, of Maxwell's equations in Sec. 13.5. 158 Looking In: Atomic and Nuclear Physics Gauss' law states that the net (outward) electric flux through any closed surface is equal to l/*« times the net charge q enclosed hy the surface : eu $ K = q or «,,/£• ds = q (13.3) Gauss' law provides a convenient way of calculating E if the charge distribution is symmetrical enough so we can easily evalu- ate the integral in F.q. (13.3). Example . A long copper tube of radius a has ;i charge of +q/l per unit length. It is surrounded by a coaxial copper tube of radius !> which earrics charge per unit length — q/l. Find the electric field (a) at a di* tanee ri from the axis where 6 > r t > a and (l>) outside this coaxial cable at, distance r : from the axis, Fig. 13.2. Fig. 13.2 in) Draw a gaussian surface which is a cylinder of radius r u coaxial with ihe cable. Since the electric field is radial, there will be no flux through the cutis of the cylinder. For a length I of cable Eq. (13.3) becomes *u/E • da = t E(2irri)l - q giving (6) When the gaussian surface is a cylinder of radius r-i, the net charge within it is (+q/l - q/l) = 0, and hence E = outside the outer conductor. 13.2 FARADAY'S LAW OF INDUCTION A changing magnetic field produces an electric field, as described in Faraday's law (Sec. 10.12). Consider a test charge (/ which Electromagnetic Radiation 159 moves around the circle in Fig. 13.3, where a uniform magnetic induction B is directed out of the paper and is increasing with time. The work done on the charge, per revolution, is the product of the emf S and the charge q. The work is also the product of the Fig. 13.3 CKorged particle q moving in a magnetic Induction 6. force qE that acts on the charge and the distance 2irr. Equating these two expressions gives £ = 2iffli (13.4) or, more generally, 8 = 6 E • dl (13.5) If this last equation is combined with £ = —d$ B /dl, Faraday's law of induction can be written (13.G) 13.3 GAUSS' LAW FOR MAGNETIC FIELDS Gauss' law for a magnetic field expresses the fact that in magne- tism there is no counterpart to the free charge q in electricity. Isolated magnetic poles do not exist. Hence the magnetic flux 'l>; ( through any closed surface must be zero: *b = <f> B • da = 13.4 AMPERE'S LAW (13.7) Ampere's law (Sec. 10.10) giving the relationship between current i and magnetic induction li can be written in circuital form as - <6 B ■ dl - i Ho T (13.8) 160 Looking In: Atomic and Nuclear Physics The line integral can be applied to any closed path near the cur- rent; symmetry usually suggests the most convenient path. Example. Find the magnetic induction Si al a distance r from a long Straight wire currying current i. Fig, 13.4 Magnetic induction S near a long, straight wire. Consider a circle of radios r centered af I lie wire for t he path of integration. Since It is tangent i<> the circle al each point .4, vectors It ami <l\ (the element of are) poinl in (lie same direction. From symmetry, It has tiie same magnitude at each point on the wire. Equation (138) becomes - 6(27rr) = ir B = **- 2irr winch is the result obtained in Sec. 10.10. Experiments show that just as a changing magnetic held induces an electric field (Faraday's law, Sec. 10.12), a changing electric field induces a magnetic field. Faraday's law for an in- duced emf G may be written i E dt (13.9) The analogous expression for the magnetic induction produced by a w hang ing electric field is - <p II • (II = eg fjl /"'■ (13.10) where the constants n» and a, are required in the mks system of units we are using. The situation expressed in Fq. (13.10) can be visualized by considering the region between the plates of a capacitor (Fig. 13.">a) which is being charged with a steady cur- rent i. The accompanying dl'I/dt produces a magnetic field: B is shown For four arbitrary points in Fig. 13.56. Electromagnetic Radiation 161 In considering the two ways of setting up a magnetic field, (I) by a changing electric field dK dt and (2) by a current i, we have assumed that there is uo current in the space considered in (1) and that no changing electric fields are present in (2). But, (a) (b) Fig, 13.5 The charging of a capacitor (a) produces a changing field dE dt, which (b) produces a magnetic field 8. in general, the contributions of both dE/dl and i must be con- sidered. Maxwell generalized Ampere's law, writing it in the form ££«.«- dt (13.11) The term t,\{d$y.'dl) has J he dimensions of current and is often called the diaplawmvtit mrri-nt. Tims, although the conduction current i is not continuous across the gap of a capacitor (because no charge is transported across the gap), there is a displacement current % D in the gap equal to the external i. 13.5 MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES We assemble here the four basic equations we have just discussed : */ f = 0E-rfl= ~ d (1 3.3) (13.7) (13.9) [ - <£ n ■ di = t^-%? + i (i3.li) m<> / dt ' We have written .Maxwell's equations in the form they have when I 162 Looking In: Atomic and Nuclear Physics no dielectric or magnetic material is present, since we are chiefly interested in radiation in "free space." Consider that the start of an electromagnetic wave occurs at the termination of a transmission line whicli is energized by an oscillating electric circuit. Figure 13.fi suggests how the electric field lines break away from an electric dipole as the charges +q and —q first approach and then recede from each other in successive time intervals of one-eighth of a period 7. An observer at /' looking toward the antenna will "see" an instantaneous electromagnetic field pattern with E down and B toward his right t = r/a r/4 3T/3 Fig. 13.6 Radiation from an oscillating electric dipole. [Adapted from S. S. A ft wood, "Efecfric and Magnetic Fields," John Wiley & Sans, Inc., New York, 1949.) (into this page). A moment later, as the wave advances, these directions will reverse. If an observer is at a relatively large distance from the source, the waves that move past hi in will be practically plane waves. For the waves shown in Fig. 13.7, B = B« sin (kx - wt) (13.12) E = Eo sin (kx - at) (13.13) Consider the rectangular prism I • I ■ dx to be fixed in space. Its trace in the z.r plane is shown in Fig. 13.86. As the wave passes over it, the magnetic flux <t?n through the rectangle will change, inducing electric fields around the rectangle. The line integral of Eq. (13.9) is dE h ; there is no contribution from the top or bottom of the rectangular path where dl and E are perpendicular. The Electromagnetic Radiation 163 magnetic flux *« for the rectangle is B(tdx). By differentiation this gives d$n , , dB W = ldx -dl From Eq. (13.9) we have dE I = -Idx (dB/dt), or dE dx dB dt (13.14) where we have changed to the d notation of partial derivatives to indicate that both B and E are functions of x and ( but that in n c y E-r<fE N i B+rfB dx (c) Fig. 13.7 to) Section of a plane electromagnetic wave troveltng to the right E parallel to i axis, 6 parallel to y axis. The part marked dx is viewed in (b) in the xz plane, and in (c) in the yx plane, evaluating dE/dx, t is assumed constant (Fig. 13.7// is an instan- taneous "snapshot"). Also, x is assumed constant in evaluating dB/dl at the particular strip in Fig. 13.7c. From Eqs. (13.12) and (13.13), kE cos (kx — oil) = o>Bn cos (kx — o>t) and we have the relation E _ CD _ F ~ & ~ c where angular frequency u> = 2ir/ wave number k = 2*/X speed c = f\ = a>/k (13.15) 164 Looking In: Atomic and Nuclear Physics The ratio of the amplitudes of the electric and magnetic com- ponents is equal to the speed c of the electromagnetic wave. In considering the trace of our rectangular prism in the yx plane, Fig. 13.7r, we see that as the wave moves by, *« changes with time and a magnetic field is induced at each point around I d.r. This induced H is the magnetic component of the electro- magnetic wave; K and li each depend on the time rate of change of the other. Since there is no conduction current £ in the space considered, the line integral of Bq. (13.10) hecomes —iilli. The flux *s through the rectangular element Idx is E{ld.t). By dif- ferentiation this gives et*a , , dE — tt- = l ax -jt M dt From Eq. (13.10) an dE aJ"** ,0 "« (13.16) where partial derivatives have been indicated for the same reasons as in Eq. (13.14). From Eqs. (13.12) and (13.13) hllu COS {kx - ut) = — enflqfi'o COS (kx — <>)t) and since c = w/k, we have U = _L = l (13.17) liu e<ijuiiw tviinC By eliminating the fields between Eqs. (13,15) and (13.17), we get c - — j— (13.18) Substitution of numerical values in this relation gives I c = v/(8.!) X 10 l2 eoulViit-m ! )(4jrl0 ' weber, ,'nt-m) - 3.0 X 10" m/sec which is the speed of light in free space. Maxwell made this calculation before it was recognized that light was electromagnetic in nature and before Herts had detected electromagnetic (radio) waves. Electromagnetic Radiation 165 The magnetic part of an electromagnetic wave is often de- scribed by the magnetic field strength //, rather than by the mag- netic induction B. For waves propagated in free space, the only case we shall consider, H = -B (13.19 Since ju« = 4r X 10 -7 weber/amp-m. // has the dimensions (weber m'-'i (amp-m/ weber), or amp/ in. The quantity EH = vVu/tu = 376.7 ohms is sometimes called the resistance of free space. 13.6 ENERGY FLOW: INTENSITY OF A WAVE Consideration of a parallel-plate capacitor can lead us to an expression of genera! validity for the* energy stored in an electric field. To charge the capacitor to a potential difference A V requires work W equal to AC AV*, where C is the capacitance. This energy is stored in the capacitor a.s potential energy. Between the plates there is an electric field E = AV/s, where « is the plate separa- tion. The capacitance is given in terms of the area A of the plates and the permittivity e u as C - t„A/s. The stored energy can then be expressed as W = £C(AF) 2 = i*,- EV = U E*As Since .4s is the volume of the space between the plates of the capacitor, we may associate the energy with the field, defining the energy density of the electric field as the energy per unit volume: u, = (k) (u E* (13.20) Similarly, consideration of the work required to establish a cur- rent in an inductive circuit suggests that the energy density for a magnetic field is „„. - ;„„//* - |B* (13.21) In a region where both electric and magnetic fields exist, the energy density is given by w = U lt E- + -^H* (13.22) 166 Looking In: Atomic and Nuclear Physics The rale at which energy is transported by a wave across unit area per unit time is called the intensity .S' of the wave. Since the wave speed is c, S - c(i«P + innH*) (13.23) By the use of Eqs. (13.17) to (13.19), the energy density can be written w = s/tofie EH = ~ EH c and the equation for the wave intensity becomes S = c- EH = EH c (I3.1M) (13.25) The directions of E, H, and the velocity of the wave are properly related if we define the intensity S as the vector cross product S= ExH (13.26) This vector is called Poynting's vector. In mks units, it gives the intensity in watts per square meter. Although we have derived this expression for the special case of a plane wave, it is a general relation which can be derived from Maxwell's equations. Example. Consider a plane monochromatic plane-polarized electro- magnetic wave (raveling horizontally northward, polarized vertically (eleeirie field intensity directed alternately up and down). The frc- y (up) i 1 x (north) Fig. 13.8 Z (east) quency is 5.0 megacycles/see. The amplitude (maximum value) of the electric field intensity E is 0/160 volt/m. (a) Give an analytical ex- pression for this wave, (6) Find the average intensity of this wave. Electromagnetic Radiation 167 The wavelength is X = c/f = 60 m. The period T = 1// = 2jt/« = 2.0 X 10 -7 sec. If coordinates are chosen as in Fig. 13.8, from Eq. (13.12) the electric field is B, = B. - E v = E B sin 2jr (H)" 060 sin (irl0 7 f - 0.105s) The amplitude //„ of the magnetic field is //„ = . p Bo = 0.060 volt/m — — , - = 1.59 X 10 < amp/m Vj«o 37 G./ ohm Since the magnet ie field is confined to the east-west direction, //, =0 //„ = //.- = 1.59 X 10-* sin (jtIO'J - 0.105s) The Poynting vector S = E X H is in the x (north) direction, and its magnitude is 5 = #o//u sin s 2v (*-*)-■ 54 X 10-«sin= (xlO' - O.lOox) The average value of sin- o over a cycle is .V, so the average intensity of this wave is S = hEoIh = 4.74 X 10-» \vatt/m» Maxwell's electromagnetic theory explained the then-known properties of light ; the experimentally measured speed of light in free space, polarization, interference, and diffraction, and the dispersion that occurs when light, passes through a medium where wave speed depends on wavelength. Extended to x rays, the theory identified them as also electromagnetic radiation, the dif- fraction of x rays by a crystal lattice being similar to the diffrac- tion of light by a ruled grating. Many seemingly diverse radia- tions were shown to be related regions of an electromagnetic spectrum of grand extent — some 80 "octaves," of which the visible spectrum comprises a little less than one octave. II. QUANTUM EFFECTS Beginning in 1900, developments took place which indicated that Maxwell's theory does not predict accurately all aspects of elec- tromagnetic radiation and absorption of energy. These develop- ments led to the quantum theory. We shall trace the quantum 168 Looking In: Atomic and Nuclear Physics hypothesis from its origin in explaining blackbody radiation, its confirming success in explaining the photoelectric effect and Compton effect, and its striking hut limited success in the Bohr model of the atom (Chap. l;">) to its merging with other hypothe- ses in wave mechanics (Chap. 16). 13.7 BLACKSODY RADIATION Any object continually emits and absorbs radiation, exchanging energy with its surroundings. If the temperature of the object is high enough, the radiation may be seen- the material glows. There is a direct relation between absorption and emission. Kirchhoff's law stales that an object which absorbs radiation of a Fig. 13,9 radiators. Good absorbers are good particular wavelength strongly is also a strong radiator at that wavelength. Consider a platinum disk in a furnace (Kg. 13.9) at thermal equilibrium. Suppose that it receives one unit of energy per unit area per unit time and that the fraction p is reflected. Then I — p is the fraction absorbed, designated a. Hut if the tem- perature of the disk is to remain constant, the disk must lose as much energy per second as it receives. The rate of emission t from the area considered must equal a. If a carbon disk is in thermal equilibrium in the same furnace, it receives the same energy per unit area per unit time as does the platinum, but it absorbs a larger fraction (1 - p') and hence must emit more, to keep T constant. Good absorbers of radiation are good emit- ters, as represented by the relative lengths of the arrows for p, p', t, and «' in Fig. 13,9. Electromagnetic Radiation 169 All materials exhibit characteristic differences in their absorp- tion of radiation in different parts of the spectrum. (The colors of many things we view are due to such selective absorption.) Hence the emission spectrum for thermal radiation at a given temperature depends on the material of the emitter. We can imagine an ideal body which absorbs all radiation incident upon it. By KirehholV's law, this body would also be the most effective emitter of thermal radiation at all wavelengths. Such an ideal absorber-emitter is culled a blackbody. Fortunately it is possible to realize blackbody conditions experimentally to any degree of approximation requited. If we form our material to make a cavity with a small opening to the outside, the hole will behave as a blackbody. Radiation which enters the hole will bounce around at the inner walls of the cavity, Fig. 13,10 Cavity radiation Is nearly blackbody radiation. gradually being absorbed. Only a tiny fraction of the radiant energy will be reflected back through the hole. Viewed from the outside, the hole is an excellent approximation to a blackbody. When the walls of a blackbody cavity are maintained at some temperature T, the interior is filled with radiation. A tiny fraction leaks out of the hole. A ray leaving the hole has, in general, undergone many reflections. At each, reflected energy is added to emitted energy until in the emerging (blackbody) radiation, the energy distribution depends only on the temperature of the cavity and not on the material of which its walls are made. The blackbody radiation can lie dispersed by a grating, and a bolometer or thermopile can be used to measure the energy radi- ated in each wavelength interval. A continuous spectrum is found; that is, radiation at all frequencies is observed. The dis- tribution of intensity per unit wavelength interval is shown in Fig. 13.11. When the temperature is increased, more energy is radiated at every frequency, and the relative increase is greater at the higher frequencies, shifting the maximum of the intensity 170 Looking In: Atomic and Nuclear Physics distribution to lower wavelengths (Fig. 13.11a) or higher fre- quencies (Fig, 13, lib). The area under the blackbody distribution curve f " S(f) df, whirl] represents the total power radiated, is found i" increase as the fourth power of the absolute temperature: P-eAT* (13.27) where P is the radiant flux from a blackbody of surface area A at absolute temperature T. This is known as the Stefau-Boltz- mann law ; the constant a has the value 5.7 X 10" 8 watt/ (m*) (°K*). Intensity I 3000 : K Intensity 2.0 4.0 (o) Wavelength, microns (6) Frequency Fig. 13.11 Blackbody radiation distribution: (a) intensity vs. wavelength, with visible region dotted, (b) intensity vs. frequency, with Ti < T. < It. 13.8 PLANCK'S LAW A number of physicists advanced theories based on classical physics to explain the distribution of energy in the continuous spectrum from a blackbody. Lord Raylcigh and Sir James Jeans assumed that radiation in a cavity has degrees of freedom which correspond to the frequencies of standing waves that are possible in the cavity and that the energy is divided equally among these different degrees of freedom. The resulting distribution law is (Rayleigh-Jeans) (1.3.28) where K» is the radiancy (power per unit area) at a wavelength X and ci and c a are empirical constants. The Rayleigh-Jeans law fits the experimental data (Fig. 13.12) only for large values of Electromagnetic Radiation 171 XT', fn fact, it leads to an "ultraviolet catastrophe" by predicting that as X becomes smaller, R\ increases without limit; the total power radiated by any body is infinitely large! Wilhelm Wien assumed that cavity radiation came from molecular oscillators among which energy was distributed with respect to frequency according to a Maxwell distribution (similar to the distribution law successfully used for molecular speeds in the kinetic theory of gases). The resulting distribution law agrees with experimental data in the short-wavelength region of the Intensity Planck's law 12 3 4 \Wgvelength, microns Fig. 13.12 Agreement of radiation formulas with Coblentz's experimental data (circles). spectrum, but predicts values of I{\ which arc too low in the region where X7' is large: th = cV-V— '" (Wien) (13.29) Max l'lanck also started with the assumption that the wallt- of a cavity radiator are made up of tiny electromagnetic oscilla- tors or resonators of molecular dimensions. He, too, used a Max- well distribution, taking e~ HlkT as the probability that an oscil- lator has energy /■.'. lie accepted the Rayleigh-Jeans calculation for the number of oscillators per unit volume in the frequency range from / to / + df. But Planck was led to make two radical assumptions: 172 Looking In: Atomic and Nuclear Physics 1. An oscillator can have only energies given by E — nhv (13.30) where c = frequency k = Planck's constant (of "action") n = an integer (now called a quantum number) The equation asserts that the. energy of the oscillator is "quantized." 2. An oscillator does nut radiate continuously (as expected on .Maxwell's theory) but only in quanta of energy, emitted when an oscillator changes from one to a lower of its quantized energy states. The quantum (or photon) radiated has energy propor- tional to the frequency of the wave: • K = hv (13.31) From these assumptions, Planck derived the distribution law R*- 2irAc 2 X 6 e*"* T - 1 _CiX-« ,,r, \r _ - (I'lanck) (13.32) where c = speed of light k = Boltzmaiui constant h = (Mi2"» X 10" w joule/sec Planck's law has been written in the second form with empirical constants Ci and c* for comparison with Eqs. (13.29) and (13.30). For a wide range of temperatures (300 to 2000° K) and a wide range of wavelengths (O.o to n2 ft), Planck's law represents the experimental data within 1 per cent. It is interesting to note that despite its initial success the quantum hypothesis was resisted by Planck himself. Conservative in nature, he tried for years to reconcile his "quantum of action" (A) with classical theory. 13.9 PHOTOELECTRICITY bight or other electromagnetic radiation falling on the surface of a metal (Fig. 13.13) can under certain circumstances liberate electrons from the metal. The number of electrons emitted pet second can he determined hy measuring the photoelectric current. The energy distribution of the electrons can he determined by applying a retarding potential and increasing it gradually until Electromagnetic Radiation 173 the stopping potential V, is found for which no electrons reach the collector. The chief features oT photoemissiou are : (1 ) There is no detect- able time la<i (> 10 ' sec'i between irradiation of an emitter and ejection of photoelectrons; (2) the number of electrons ejected per second is proportions! I to the intensity of radiation, at a given frequency; (3) the photoelectrons have energies ranging from zero up to a definite maximum, which is proportional to the fre- quency of the radiation and independent of its intensity; (4) for each material there is a threshold frequency v K below which no photoelectrons are ejected. * Light source Fig. 13.13 Apparatus for photoelectric effect. These characteristics of photoemissiou cannot be explained by Maxwell's theory of electromagnetic radiation. In MM)"> Kinslein made the assumption that light of frequency c can give energy to the elections in f he metal only in quanta of energy hv. Fit her an electron absorbs one of these quanta, or it docs not. If it is given energy hv, an electron may use an amount of energy v> in escaping from the metal, where it has negative potential energy, into the vacuum, where it has zero potential energy. The quantity w is called the work function of the surface. The maxi- mum kinetic energy which the electron can have when it leaves the surface is therefore /■: *. NIEIJS = hv — w (13.33) This is called Ivinstein's photoelectric equation. It explains the linear relationship E i: = an + h shown in Fig. 13.1-1: The slope a measured from the graph agrees with the value of Planck's con- stant A; the negative intercept b is identified with the work func- 174 Looking In: Atomic and Nuclear Physics lion id of the metal. The intercept on the frequency axis is the minimum frequency of light that will liberate electrons from the particular metal. At this threshold frequency vt, the photon delivers just enough energy to enable the electron to get out of the metal (with E k = 0): hvo = w (13.34) From Eq. (13.33), #*.„„,* is independent of the intensity of illumi- nation, in agreement with experiment. The term photoelectricity includes several distinct phenomena. In the external photoelectric effect (photoemission), electrons *■!, M. Fig. 13.14 Dependence of maximum energy of photoelectrom on frequency. are ejected from a solid (or liquid) surface into a surrounding vacuum. Photomultiplier tubes use this effect. Electrons and ions may he produced in a gas by ptiotoionista- tion. The ionization chambers used to detect x rays utilize this effect. Conduction electrons and positive "holes" which remain inside a solid may be responsible for either photoconduction or a photovoltaic effect. Photoconduction is a decrease in resistivity under the influence of radiation. It is used in television camera tubes and in control devices where an external battery furnishes the electric power. The photovoltaic cell is a device for converting radiation into electrical power. Kadiation acting on two dissimilar layers in the cell gives rise to an emf in much the same way a voltage is produced when Cu and Zn plates are dipped in acid in a Electromagnetic Radiation 175 voltaic cell. Photovoltaic cells are used in photographic exposure meters and in solar batteries. There is also an internal photoelectric process, within an atom, called the Auger effect, or autoionization. An x-ray quan- tum may be absorbed within the same atom from which it origi- nates, with the ejection of one of its electrons. The net effect is thai the atom adjusts from an excited level to a lower-energy level, with the emission of an electron. Finally, there is an inverse photoelectric effect in which an electron is absorbed by a solid and a photon emerges. The photoelectric effect gives strong support to Planck's hypothesis that light of frequency v can be emitted or absorbed only in packets of energy hv. The citation which accompanied the award of the Xobel Prize to Einstein stated that it was for "his attainments in mathematical physics and especially for his dis- covery of the law of the photoelectric effect." 13.10 THE CONTINUOUS X RAY SPECTRUM X rays are electromagnetic waves of very short wavelength, about 10"" to I0~" m. In an x-ray tube (Fig. 13.1")) a battery li Fig, 13.15 An x-roy tube. heats a tungsten filament C so that it emits electrons. A potential difference of several thousand volts between cathode (' and target T accelerates the electrons. The fast-moving electrons are quickly decelerated when they strike the metal target. Most of their energy is converted into heat by collisions with atoms of the target. But as the electrons are decelerated, they are expected to 176 Looking In: Atomic and Nuclear Physics radiate, according to Maxwell's electromagnetic theory. The Gorman term Itrrmxxtniltliniu is used for this "slowing-down radiation." The radiation is emitted in all directions. When one examines the beam of x rays emerging from the hole in a lead shield, one finds a continuous distribution of frequencies up to a certain maximum. This maximum frequency depends on the potential difference at which the tube is operated: v,„„ x /AV = a constant for a wide range of voltages. The high-frequency limit in the continuous x-ray spectrum is difficult to explain classically. It is easily clarified by the photon hypothesis. An electron may suffer numerous decelerations as it encounters various atoms in the target. ICach time a photon is produced, whose energy hi> is equal to the decrease in kinetic en- ergy A A'* of the electron. Clearly, the highest-frequency photon that can be produced is that which results from the complete. conversion of the electron's kinetic energy into a single photon. Since electrons arrive at the target with energy e Al". hv mnx = e AV (13.35) From this Duane and Hunt law, Eq. (13.35), r'h may be deter- mined from the sharp cutoff of the x-ray intensity versus fre- quency curve at p„ k , x . There is good agreement with the ratio e/h determined in other ways. 13.11 THE COMPTON EFFECT Another even more direct confirmation of the photon hypothesis came about |<)23 in A. H. Compton's explanation of properties of scattered x rays. Compton allowed a beam of monochromatic x rays to fall on a hlock of scattering material such as carbon. The scattered radiation was examined in an x-ray spectrometer (an instrument which uses » crystal and an ionization chamber to measure the wavelength of x rays incident on it J. Spectrometer /-\ C X Fig. 13,16 Apparatus for observing Compton scattering of x roys. Electromagnetic Radiation 177 According to classical theory, scattered radiation should have the same frequency as the incident radiation. Compton found such unmodified radiation, but in addition he found a scattered wavelength X' greater than that of the incident beam. The shift In wavelength X' - X was found to increase as the angle 8 at which the scattering was observed was increased (Fig. 13.17). The scattering of x rays with increase in wavelength is called the Fig. 13.17 Wavelength shift in Compton scattering of x rays. Wavelength, X — *" 178 Looking In: Atomic and Nuclear Physics Campion, effect. The. plioton hypothesis provides a straightforward explanation. We describe the Compton scattering as an elastic collision of a photon with a free electron which is at rest before the collision (Fig. 13.18). We ascribe to the photon the "equivalent mass" hvfe* (Chap. 14), and to this mass we attribute linear momentum hv/c. The conservation of momentum may be stated in two Photon * Before After Fig. 13.18 Compton scattering of a photon. Electron equations, since momentum is a vector quantity and the law of conservation applies to each of the components: kv hv „ , X component: — = — cos 8 + me cos # c c hv' . y component: = — sin $ — mv sin 4> (13.30) (13.37) where v' is the frequency of the scattered photon and me is the momentum of the recoil electron. Conservation of energy requires that hv = hv' 4- A* (13.38) where K k is the final kinetic energy of the electron. Solution of these three equations provides an expression for the wavelength shift X' - X = -- (1 - cos 9) mr (13.39) which agrees with experimental data. The unmodified radiation is interpreted as due to photons scattered by electrons strongly bound in atoms. Electromagnetic Radiation 179 Example. What is the change in wavelength <>r x rays Com|ii on-scattered in the backward direction (6 = 180°)? X' -X = 6.625 X 10 3t joule sec 9.1 X 10 3l kg (3.0 X IIP in sci") = 0.048 X IO" 1 * m = 0.0484 A (1 - cos 180°) 13.12 WAVE-PARTICLE DUALITY: PROBABILITY We have discussed two theories of electromagnetic radiation. The classical theory says iliat radiant energy flows continuously as a wave. The wave theory gives a satisfying explanation of interference, diffract ion. and polnrixa! ion experiments. The quan- tum theory says that radiant energy is exchanged in quanta of amount Ac, whose value depends on the frequency c of the light. This photon theory gives a satisfactory interpretation of many experiments in atomic physics (blackbody radiation, photo- electric effect, the frequency limit in a continuous x-ray spectrum, the Compton effect, and the line spectra characteristic of elements). In some ways these two theories are mutually contradictory. The wave theory says that the photoelectric effect should show a time lag when the light source has a very low intensity. The photon theory when used to explain a single-slit diffraction pat- tern would have to assert that these particles arriving at certain points on the screen would "cancel" each other. (How?) Although each theory works well for its own experiments, something has to "give" when we try to put the two theories together. A resolution of this tlillieulty was suggested by a novel idea proposed by Louis de Broglie, in his 1'h.D. thesis in 1924. From consideration of relativity theory (Chap. 14), he deduced that all particles must have a wave nature, just as light has a wave nature. The intensity of the particle wave at any given point (or the square of the wave amplitude) is interpreted as proportional to the probability of finding the particle at that point. The de Broglie relationship for the wavelength X of a matter wave is X = * V (13.40) where p is the momentum (mv) of the particle and h is Planck's constant. rr 1 * 180 Looking In: Atomic and Nuclear Physics The exploration of wave-particle duality was continued in M)28 by Max Horn. as follows. Kuergy is not distributed continu- ously throughout an electromagnetic wave: the energy is carried by the photons. The intensity of the wave (which the classical theory defines as energy flow) at a point in space is really a meas- ure of the probability of finding a photon there. The classical wave has become a sort of guide for the individual quanta of energy. We have resolved the wave-particle dilemma, but at the cost of admitting that laws of chance govern the motion of micro- CDBEAEBDC Fig. 13.19 Diffraction at a single slit. Fig. 13.20 Double-slit diffraction. scopic particles. If we photograph a single-slit diffraction pattern (Fig. 13,19), the relative intensities tell us that many photons struck the plate in the region .4, a fair number in li, some in C t etc. Very few hit near I) or /■.'. Suppose, now, we perforin the experiment with one photon. We can predict that it has a high probability of hitting near .1. a fair chance for li. less for ('. very little chance of hitting near i) or K, Hut we 08JQ prediel to which point the photon will actually go. Consider an experiment in which a beam of electrons falls on two slits (Fig. 13.20). The electron distribution at I 1 predicted Electromagnetic Radiation 181 by classical theory is shown by the dashed line. The distribution actually observed (solid line) is that predicted by considering interference of the tie Hroglie waves. Now consider one electron shot at a time. According to this wave picture, each electron is represented by a single wave packet which divides equally be- tween the two slits. Vet if we place a particle detector at slii .1, we never observe half an electron: we find cither a whole particle or no particle. It is intriguing to try to devise an experiment that would reveal the slit used by individual electrons, without de- stroying the interference pattern. No one has succeeded. If a detector is placed at A, the interference pattern smooths out; the classical result is obtained. 13.13 THE UNCERTAINTY PRINCIPLE A consequence of quantum theory is that one cannot determine simultaneously the exact position and velocity (or momentum) Fig. 13.21 Supermicroscope. o/vw of any particle. As an example, assume that the exact momentum of a particle is known. Then it has a definite wavelength X = h/p and is a continuous plane wave of uniform "intensity." It is equally probable to find the particle anywhere in space. At the other extreme, assume that we have located the particle within a very small region of space. Then its wave function is a short packet that does not have any unique X. Hence the momentum is fuzzy. The uncertainty principle predicts that in general we cannot make a measurement on a system without disturbing it. for example, suppose we try to "view" an electron with an (imagi- nary) supermicroscope .1/ (Fig. 13.21) to determine its position x and momentum p. We may borrow an expression from optics which says that the smallest displacement A.v the instrument can 182 Looking In: Atomic and Nuclear Physics defect depends on the wavelength of the light and the half-angle a subtended by the objective lens: Ax = X/(a sin a). We "view" the elect ion by light which (-liters the microscope anywhere within angle 'la. This radiation, scattered by the electron, makes a con- tribution to the electron's momentum which is unknown by Apz = p sin a — ^ sin a If we write Ax as the uncertainly in position of the electron and Ap* as the uncertainty in its momentum, combining the last two equations gives 1 / X \ (k . \ h Ax Ap x = I -. — 1 [ :- sin « J = - a \sm a/ \\ / a (13.41] which shows that as we increase the precision of our measurement of 3% the value of p becomes subject to greater uncertainty. (The foregoing is offered merely to amplify the statement that wc can- not make a measurement on a system without disturbing it. The numerical value of a depends on the criterion used for resolving power.) Werner Ileisenberg formulated the uncertainty principle in 1927 showing that from Sehrodinger's equation (Chap. Hi) /i/4ir is the lower limit of the product of simultaneously measured value of a particle's position and momentum: Ax Ap > -r- (\:i.m Mere Ar and A/; are defined as vm- deviations. There is an uncer- tainty relation only between certain pairs of variables, those which are "canonically conjugate variables."* There are uncer- tainty relations, for instance, between position and momentum (discussed above), angular momentum and angle, and energy and time. We have seen that quantum theory is significantly different from classical theory in dealing with the interactions and struc- * See Condon and Odtshaw (wis.), "Handbook of Physics," cinq). I>, McGraw-Hill Book Company, Inc., New York, 19oX. Electromagnetic Radiation 183 tore of small particles, IT the quantum theory is correct, as we think, there is no hope of understanding the elementary structure of matter (atomic and nuclear physics) from the viewpoint of classical physics. In the following chapters we shall use the ideas of quantum theory. It will be interesting to see, however, that there is a region between macroscopic and microscopic physics where the laws of classical and quantum physics smoothly overlap (correspondence principle). The secret of education lies in respecting the pupil. R. W. Emerson The most essential characteristic of scientific technic is that it proceeds from experiment, not from tradition. Bertrand Russell The most brilliant discoveries in theoretical physics are not discoveries of new laws, but of terms in which the law can be discovered. Michael Roberts and E. R, Thomas Relativity Wonderland 185 14 tivistic mechanics. The insight it gives into the binding energy of nuclei and the liberation of nuclear energy will he our cbier interest. We shall look, also, at what relativity says about simul- taneity of events, time dilatation, and the aging of voyagers in spaceships. Relativity Wonderland The supreme task of the physicist is to arrive at those universal elemen- tary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them. A. Einstein 14.1 NEWTONIAN RELATIVITY In elementary experiments in mechanics we recognize that trans- lator? motion can be measured only as motion relative to other material bodies, such as the workbench or the earth. Measure- ment of a speed involves measurement of both distance and time. vt S' Fig. 1 4.1 Reference system S' moves with constant velocity v in x direction relative to reference system S. Relativity and quantum mechanics are two great theories of twentieth-century physics which have modified in remarkable ways our ideas of the physical universe. For bodies traveling at speeds close to the speed of light, Newtonian mechanics is replaced by rclativistic mechanics. The relativity theory of the physical meaning of space and time makes some simple predictions of great importance. (I) The mass of a particle is shown to be vari- able, depending on the speed of the particle; (2) it is impossible for any particle to have a speed greater than the speed of lighl ; (3) mass and energy are interconvertible. In this chapter we shall consider the evidence which leads to the formulation of rela- 184 Up to now we have perhaps intuitively regarded time as a unique variable, quite distinct from, say, space, energy, or the* behavior of material things. \Ye might agree with Newton that "Absolute, true, and mathematical time, of itself, and by its own nature. Hows uniformly on, without regard to anything external." It is helpful to formulate the kind of relativity implied by these ideas for later comparison with the new relativity. Consider a material reference body and some sort of timing device (the rotating earth, or a crystal oscillator) to constitute a space-time system of reference for making measurements to locate particles or to describe events. Now suppose a second system of reference S' (Fig. 14.1) to be in uniform motion with respect to the first reference system >S\ along the common line of i heir x axes. Let the velocity of S' relative to S be v. Let us agree to reckon time from the instant at which the two origins of coord i- 186 Looking In: Atomic and Nuclear Physics nates and 0' momentarily coincide. At any later time, the co- ordinates of 0' measured in system S will be x = vt, y = 0, z = 0. An event which occurs at coordinates x,i/,z and time / in system A' will, according to Xcwlon, have coordinates in system S' given by v* - y X = z f = t (14.1) 14.2 THE AETHER The wave properties of light were, demonstrated by Young, Fresnel, and others during the first part of the nineteenth century and were explained in Maxwell's brilliant theory of electromag- netic radiation (Chap. 13), It was difficult for scientists of the nineteenth century, as for us, to conceive of a wave motion without a material medium to transmit its vibrations. So they invented a medium called the aether for the propagation of light. The aether was thought to pervade all space, as well as trans- parent material bodies. The assumed existence of the aether sug- gested two interesting consequences worthy of experimental check, (I) Light waves should travel with a definite speed (c = 3.0 X 10" m see in "empty" space) with respect to the aether itself. Then the apparent speed of light relative to a mate- rial body moving through the aether should be different from c and should depend on the speed of the body. (2) An "absolute" velocity of the earth or any other body should be ascertainable from measurements on tight waves transmitted through the aether. 14.3 MICHELSON'S INTERFEROMETER An experiment designed to detect the motion of the earth relative to the aether would require very sensitive apparatus, for the orbi- tal speed of I he earth is only about III ' the speed of the light signals that would be used in the measurement. With this prob- lem in mind, Miehelson devised an interferometer, an instrument in which interference patterns produced by two light beams are used to reveal differences in the optical paths of the beams. When the optical paths (Fig. 14.2) happen to he equal, beams 1 Relativity Wonderland 187 and 2 will arrive at E in phase and produce a bright field, by con- suuetive interference. As distance .1.1/ is increased X I by moving mirror .1/, the optical path for ray 1 is lengthened by X/2, and destructive interference of rays 1 and 2 at li gives a dark field. r c\ 3-w ~G\ M Source Fig, 14,2 Michelson's i nf erf ero meter. Mirrors .1/ and .1/' are set nearly but not quite perpendicular to produce a field crossed by alternate bright and dark interference fringes (Fig. 14.8). These are counted as they move past a refer- ence mark R, For each fringe that passes the mark, the optical path has changed by one wavelength. This change might he pro- duced by moving .1/ a half wavelength. However, the change might also he produced by a change in the speed of light in beam I (on the substitution, for example, of a gas of different index of refraction for the air in that one beam). Fig. 14.3 Fringes and a reference mark. 188 Looking In: Atomic and Nuclear Physics 14.4 THE MICHELSON -MORLEY EXPERIMENT Assume that the earth travels through stationary aether with a speed i> and that light has a speed c in the aether. Consider a Mieheison interferometer arranged so that one of its two equal arms is parallel to the earth's velocity {Fig. 14.4). Then the. times required for the light beams to travel the distances AM A and AM' A will be unequal. The speed of a beam traveling from .1 to .1/ is c — t" relative to the interferometer. On the return from .1/ M' i A 4—W£ i M n HI u Fig. 14.4 Light poths in moving interferometer and velocity vector diagram. to A, the speed of the beam relative to the interferometer is c + v. The time for the round trip AM A is thus c — V C + V 2w (14.2) Since r is small compared with c, we may use the binomial theorem to obtain the approximation 1 — rye 2 c \ c* / (1-U) A wave front leaving A toward mirror .1/' will be returned, according to Huygen's principle, but only after A has moved to a new position .]'. The component of the velocity of light in the direction perpendicular to the motion of the interferometer is VV 1 — v i . The time for the round trip AM' A is k « 2s 2»/ vV - v* Vi !n /; _ 2* / Iff* 2 ri 1+3^ + ) (14,1) Relativity Wonderland 189 Waves which are in phase when they reach A from the mono- chromatic source will differ in phase when they return to .4 after reflection, because of the time difference: Ai = ft — h = si- (14. o If the interferometer is rotated 90°, paths t and 2 will have their roles interchanged and the total retardation will be 2sp ! The number of fringes passing the reference mark should be N = path difference _cA/ = c2st>* _ 2siP wavelength X Xc* c 2 X (14.6) To estimate the magnitude of fringe shift to be expected, we may assume that the earth's velocity through the aether is the same as its orbital velocity, about HO km sec. By using multiple reflections, .Mieheison and Morley attained an effective path s of 10 m (Fig. 14.5). For light of wavelength 5,000 A we should then estimate a maximum fringe shift of _ 2 X 10m(3 X lOWsec)' _ f . ( 4 ?) N ~ OTX 10«m/»»)»(5.0 X 10- T m) U * lr " lge U4 ' J A fringe shift of this amount is readily detectable with the apparatus. It should then be possible to measure the fringe shift and from it compute the velocity of the earth relative to the aether, that is, the absolute velocity of the earth. Surprisingly. Mieheison and Morley found no fringe shift when the interferometer was rotated in a pool of mercury. It- appeared that optical experiments cannot detect motion of the earth relative to the aether. Mieheison and Morley reported their results in 1887. No subsequent experimental evidence contradicts them. Some linger- ing doubts were laid to rest in a review article published in the Reviews of Modern Physics (pages 107- 178) in 1955. Several attempted explanations for the apparent impossibility of measuring the earth's absolute motion failed to gain acceptance when they either did violence to established theory, disagreed with known astronomical data, or introduced too many special hypotheses. m Fig. 14,5 Michelson interferometer designed to detect "absolute motion" of the earth, (o) Interferometer was mounted on a stone, ond floated in mercury to damp vibrations and to permit rotation, (b) Multiple reflection of beams gave an effective path length of 10 m. 190 Relativity Wonderland 191 14.5 POSTULATES OF THE SPECIAL THEORY OF RELATIVITY Consider several physicists in a completely enclosed elevator or railroad car, moving with constant velocity relative to earth. Could these people detect and measure the velocity of their enclosure from observations made inside, with pendulum, spring balance, etc? Proceeding from considerations such as these, Henri Poincard in the period 1 SiM) to 1904 developed the hypothesis that it is impossible to determine absolute motions of a body or of a refer- ence system by any dynamical, electromagnetic, or optical means. Measurement of the velocity of bodies relative to a stationary net Iter seemed to Iks the best device classical physics could offer for determination of "absolute" motion. The negative result of the Michelson-Morley experiment was interpreted by Einstein as indicating that only relative velocities can be measured. Con- sequently, the general laws of physics must be independent of the velocity of the particular reference system of coordinates used to state them, otherwise it would be possible to ascribe some abso- lute meaning to different velocities. The special or restricted relativity theory of 1905 was limited to consideration of reference systems moving at a constant velocity with respect to each other {Fig. 14.1). Einstein based his theory on two postulates: 1. The laws of physical phenomena are the same when stated in terms of either of two reference systems moving at constant velocity relative to each other {and can involve no reference to motion through an aether). 2. The velocity of light in free space is the same for all observers and is independent of the velocity of the light source relative to the observer, {The "general" theory of relativity, 1 9 Hi, is Einstein's theory of gravitation and will not be considered here.) Suppose person A. at rest in a laboratory, assigns to every event which he observes a position (j-.i/a) relative to a particular origin fixed in his laboratory and a time J as indicated by a clock at rest in his laboratory. Now let person B move through A's laboratory with speed u in A's positive x direction. Let person 192 Looking In: Atomic and Nuclear Physics B measure positions relative to an origin moving with him and times with a clock (just like A's clock) also moving with him. Then to each event B will assign a position (x',y',z') and a time I'. Assume that the clocks are synchronized to read I = (' = when the (x'y'z') axis momentarily coincides with the (xi/z) axis. The relations which connect the distance and time intervals between two events as measured from the two inertial reference frames are x = — V = y' z = z' x' + lit' == X' = U' - !J Z' = 2 X — ut » _ t_— (u/c*)x Vi - 1* 1 /** (14.8) These transformation equations were developed by Voigt (1887) and Lorentz (1904) in exploring the aether hypothesis. But Einstein showed that the transformations satisfied his rela- tivity hypothesis that the speed of light will be the same in each coordinate system. Example. Show that light has speed c in both the 8 and S' coordinate systems. Suppose that the light starts from x = 0, y = 0, t = at J = and moves in the positive x direction. It will arrive ill the point x = A' at the time X/e, time its speed through the laboratory is ,-.. Person B will observe the light to arrive at the point x , m X - u(X/c) Vl - u7c* at the time t , m X/e - (uM )X Vl - m*/c* The speed of light in the S' coordinate system is thus x' X - (u/c)X V = -7 = I' X/e - (u/c*)X = r Relativity Wonderland 193 transport energy from one point to another with a speed exceeding the speed of li<iht. Several relations of particular interest will now be discussed to illustrate the meaning of space and time variables. 14.6 VELOCITIES NEVER ADD TO MORE THAN c Suppose that our two observers in coordinate systems S and S' both observe an object which Hies past in the x direct ion. Observer B measures the speed of the object relative to him as v' = dx'/dt'. If we express v' in terms of the coordinates of the laboratory observer A, we find dt' d[(x - mQ / V'1 - u */c*] dx -udl d[[t - (*/c*)*]/Vl - »Vc*l di ~ (*/<&* -r^ws (l4! » where in the last step numerator and denominator were divided hy dt and dx/dt = » was written for the speed of the object in the laboratory. Thus we have »' + « V = V — u or v = 1 + uv'/c* (14.10) 1 — uv/c 1 The speed v relative to the laboratory is not, as we might have expected, exactly equal to the speed v' relative to B plus the speed u of B relative to the laboratory. Example. While observer B is moving through the laboratory with speed u = 0.90c, a flying object passes him with a speed which he meas- ures as v' = 0.90c. What is the speed of the flying object relative to the laboratory? v = V ' + « 0.90r + 0.90c I.SOc 1 + w'/c* 1 + (0.90c) (0.90c)/c ! 1 + 0.81 = 0.994c In other words, if a car were traveling at speed 0.90c, you would have to drive at a speed of only 0.994c to pass it with a relative speed of 0.90r! In a mathematical sense, the principle of relativity is that the equations of physical phenomena must be invariant in form under Lorentz transformations. The basic physical assumption of rela- tivity is that no mechanical or electromagnetic influence can 14.7 WHAT DOES "SIMULTANEOUS" MEAN? Einstein pointed out by the following railroad story that man cannot assume that his sense of "now" applies to all parts of the 194 Looking In: Atomic and Nuclear Physics universe. He pictured a straight section of track with an observe? seated on an embankment beside it. During a thunderstorm, two lightning bolts strike the track simultaneously, at points Xi and .)■•:. Kiustein asks: What do we mean by "simultaneously"? x i A Fig. 14.6 My time is not necessarily your time. Assume that the observer is seated midway between r, and .<-._>, Assume that he has arranged mirrors so he can see x t and z» at the same time without moving his eyes. Then If the reflections of the lightning flashes are seen in the mirrors at precisely the same instant, the flashes may be regarded as simultaneous, by ob- server A. Now assume that a train speeds along the track and that observer B on the train sits in an observation dome, with an arrangement of mirrors for viewing points x x and x->. It happens that observer B finds himself directly opposite A when lightning strikes the rails at a and x-,. Will the flashes appear simultaneous to B? No, for if his train is moving from x-, toward .c,, then the flash at .10 will be reflected in his mirrors a fraction of a second later than the flash in .r,. (In the limiting case with a train travel- ing at speed e, B would never see light from x«.) Whatever the speed of the train, the observer B on it will always say that the lightning Hash ahead of him has struck the track first. In generalizing, we are forced to admit that two events which occur at different places may be simultaneous for one observer and not simultaneous for another. We cannot assume that a single time scale (( = (') can be used with any and all coordinate systems. 14.8 THE FITZGERALD-LORENTZ CONTRACTION To explain the null result of the Michelson-Morley experiment, Fitzgerald in 1893 arbitrarily assumed a contraction of the arm of the interferometer in the direction of motion of the apparatus. Relativity Wonderland 195 The special theory of relativity predicts the same contraction but ascribes it to the relative motion of the body and the observer. Consider a material object in coordinate system S' whose surface may be defined by the relation 4>{x' ,tf ,z') = 0. Then, by the Lorentz transformation, the form of the. surface as viewed in coordinate system S is In particular, suppose that a spherical surface of radius a is described in system S' by (Y) 1 + Cv') 2 + {*')" ' a" = <>■ Tbfe appears in system S to be a moving ellipsoid ( x r*J + K + 3-1 whose semiaxes are (a \/\ - u- c-,a.a). The surface undergoes contraction in the direction of motion in the ratio y/\ — u s /c 2 : 1. 14.9 TIME DILATATION: THE CLOCK PARADOX Consider now ihreflVel of relative motion on a flock. Two events occur at a point in coordinate system .S": one at time t\, the other at a later time 4 T <> an observer in S these events take place at different points in space, (.ri,//,s) and Oj.(/,z), as well as at differ- ent times, such that {x» - xi) = u(t t - h). Prom the Lorentz transformations tt - h = Thus the sequence in time of the two events is the same, but &t appears longer for the observer in S than for the observer in .S". This is interpreted as meaning that a moving clock appears to run at a slower rate than does an identical clock at rest, in the ratio Vl - «7c°-:l. The imminence of space travel has revived interest in the "clock paradox" or "twin paradox." One of two identical twins leaves his brother on earth and voyages at high speed into dis- tant space. On his return, he finds that his brother has grown 196 Looking In: Atomic and Nuclear Physics much older than he, because of time dilatation in the spaceship. Superficially, this is a paradox, for it challenges "common sense." Also, it seems to contradict the assertion of special relativity that in describing physical events all observers are equivalent; none has a preferred or absolute reference system. The aging or clock effect seems to provide a way of distinguishing among observers. But, relativity asserts the equivalence of observers in inertial systems, and since one of the twins accelerated at the start of his space trip and again when he altered course to return, he did not view his brother from the same inertial system before and after the trip. So there is no paradox. The intriguing question remains: Did the stay-at-home brother grow older faster? Yes. In his 190") paper "On the Elec- trodynamics of Moving Bodies," Einstein wrote, If at the points A and B there arc I ho stationary clocks which, viewed by a stationary observer, arc synchronous, and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the (.wo clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by tv 3 /2c- (up to magnitudes of fourth and higher orders), ( being the time required for the journey from A to B. It is at once apparent that this result still holds if the clock moves from A to B in any polygonal line, and also when the points A and B coincide. Bergman n has suggested the following elucidation of the clock effect, tig. I 1.7. 9"-^= — I" Fig. 14,7 Clock paradox. Observer A arranges for periodic light signals to go from lamp L to mirror .1/ and back (a kind of optical clock). Light travels a distance 2/) for each LM L circuit. Observer B is moving with constant speed u at right angles to the line LM. For him, Relativity Wonderland 197 the same light signal travels the larger distance 2D'. If observer B set up a similar experiment in his coordinate system S', his light signals would complete their round trips in shorter times than noted by observer A. The discrepancies arise because the two observers do not agree on which of two distant events {com- pletion of the nth round trip by either light signal) takes place first. Now let observer B suddenly reverse his velocity (u[). He is now in a different Lorentz frame. (He accelerated.) His notions of simultaneity have changed. Observer A sees B coming toward him, with B's light signals arriving slower than his own. When they meet, A's signals have completed a larger number of LM L circuits than have B's signals. Observer A has aged more than B. Example. What, will he the difference in I he rates of two identical clocks, one of which is on a spaceship moving at 300 mi sec relative u> the other? u = 300 mi/sec = 5.25 X 10 s - m/scc c = 3.0 X 10 a m/sec Relative change in rate f. u* f, S^iTx 10" / - — - — = 0.002 per cent, approx. 10"« Experimental detection of time dilatation was achieved by Ives and Stilwell (1938) on viewing the spectral lines of hydrogen atoms which were given a high speed directed away from the spec- troscope. An arrangement was used to distinguish relativity effects from Doppler effects. Light from the atoms fell on the spectrograph slit directly, and also after reflection in a mirror set at some distance and normal to the velocity of the atoms. Owing to the Doppler effect, each spectrum line was split into two fre- quencies. Then light from hydrogen atoms at rest was viewed with the same spectrograph. This gave lines slightly displaced, in frequency, from the middle, of the Doppler pairs, in amount predicted by relativity, .Measurements of the lifetimes of mesons have been used to cheek relativity predictions. The mean life of fi mesons (about 198 Looking In: Atomic and Nuclear Physics '2 X 10"* sec) has been found to depend on I heir speed roughly in (lie way predicted by relativity. 14.10 MASS AND ENERGY Two results of relativity theory which are of especial importance in atomic physics are (1) tlie variation of the mass of a particle with its speed and (2) the equivalence of mass and energy. Experiments have been performed, first by Bucherer in 1909, on the deflection by a magnetic field of electrons whose speeds are not small compared with the speed of light. The acceleration may be determined from the radius of curvature of the path a = v*/R. The force producing this acceleration is the magnetic side thrust on the electron / = Bev. It is found that for high- speed electrons Newton's law in the form/ = ma is not satisfied. But Newton's law written in the form / = d(mv)/dt is satisfied, provided we assume that the mass m of the particle depends on its speed. It is found necessary to assume that a particle which has mass m u when at rest has a mass m = m-o y/\ - i.'Ve* (14.11) when moving with speed v. The quantity wio is called the rest mass. When v « c, m = m-u. Variation of mass with speed is accepted in relativity theory as requisite for the conservation of momentum, which remains a basic principle of mechanics. In order to have the total momen- tum of an isolated system remain constant, the momentum of a panicle is delined as p = mv = Met) Vl - »Ve* (14.12) Table 1 4. 1 shows for various ratios of v/c the kinetic energy of an electron, the ratio vi/m , and the product BR, from which one may get the radius R of the path in a magnetic induction B of given value. Looking at the table, one might say that in problem solving to slide-rule accuracy, one can neglect relativity variation of mass for bodies having speeds less than 0. 1 l he speed of light. Relativity Wonderland 199 Table 14.1 Date on electrons V c Energy, ev ffl fllfl 8R, X10 • weber/m 0. 1.00000 0,0100 25.54 1.00005 17.0 0.0200 102.2 1.00020 34.06 0.0500 638.5 1.00125 85.0 0.100 2,575 1 .00504 171.3 0.200 10,530 1 .02062 347.8 0.500 79,030 1.1547 983.6 0.600 127,700 1.25000 1,278 0.700 204,300 1.4002 1,669 0.800 340,500 1.6666 2,272 0.900 661,000 2.2941 3,517 0.990 3,110,000 7.0888 11,960 The kinetic energy of an object having speed r is equal to the energy required to accelerate it from rest to the final speed v. E h - //-'(cos $) ds = jT dx (14.13) But now we must use in place of/ — ma for Xewtou's second law, d me F- givmg dt y/l - B */ c » ft- [* »g_ d X = m f r vd( ^=) J dt y/\ - r * J3 JO VVl - W/ This may be integrated by parts using the standard form /« dv = uv — $v du to obtain ft = ffloiM — 7= - 1 J ft = »ic 2 — muc 2 (14.14) (14.ir>) This expression replaces the classical formula hn^ for kinetic energy when v is comparable with c. The equation for kinetic energy, Kk = (m — m.n}c' i , says that when we speed up a particle, the increase in energy is propor- tional to the increase in mass of the particle: &S = c-(Am) (14.10) 200 Looking In: Atomic and Nuclear Physics We can identify c- times the relativist ie mass of the particle with the total energy K of the particle: E = mc * = A* + Wo c 2 (14.17) Total energy = kinetic energy + rest energy Kinstc.in's famous relation IC = mc % states that mass and energy are different aspects of the same thing. It tells us tiie rate at which one may he converted into the other. Example. Find the energy equivalent of 1 gm of coal (or any other substance). E = me 2 = 0.001 kg (3.0 X 10* m/sec,} 1 = 9 X 10 13 joules = 25,000 niogawutt-lir Only a liny fraction of this amount of energy is released in the burning of I Km of coal: tin- combustion products have a mass only sJightly less than 1 Kin. In nuclear reactors, a somewhat larger percentage conversion takes place, but it is still a small fraction. 14.11 NUCLEAR BINDING ENERGY Mass spectrograph measurements show that the mass of any stable isotope is less than the sum of the masses of its constituent protons, neutrons, and electrons. Kinsfein's mass-energy relation suggests thai the mass discrepancy might account for the energy needed to hold a nucleus together, against the dispersive forces exerted by the protons on each other owing to their positive charges. The mass of the constituent particles for nucleus zX. A is the sum of Z proton masses and (A-Z) neutron masses. The mass defect Ant is then calculated from Am = Zm.fi + (A — Z)m„ — M z,A (14.18) where ma = 1.00814o amu, mass of the hydrogen atom m„ = 1.00898b' amu, mass of the neutron M x , = mass of the neutral atom of atomic number Z and and atomic mass number .1 From A/i = (Awi)c 3 one can calculate that I amu is equivalent to Oil I Mev (million electron volts). Binding energy is primarily a property of the nucleus. Yet in the equation above we have used data for neutral atoms >»u and (14.19) Relativity Wonderland 201 Mx..\ which of course include electrons. We justify this procedure by the following facts: (1) If a nuclear reaction is written in terms of the symbols for the corresponding atoms, the number of elec- trons on one side of the equation generally cancels the number of electrons on the other side. (2) The minute changes in mass which may accompany the formation of an atom from its ion and electron(s) is negligible. (3) The mass data from mass spectro- graph experiments are always tabulated in terms of neutral atoms (e.g., Na) even though deflection measurements must be made on ions (e.g., Na ++ ). It is to avoid the trouble of specifying each time the degree of ionization that the experimenter adds to his experi- mental value for the mass of the ion the proper number of electron masses and reports as the isotope mass the* computed mass of the neutral atom. The binding energy per nucleoli is defined as the binding energy divided by the number of nuclear particles: Binding energy _ Am <■- Xucleon .1 It is this value which is significant in comparing the stability of two different isotopes. 14.12 RELATIVITY AND SPACE TRAVEL Rockets for space exploration require highly efficient sources of thrust and large amounts of electric power. These requirements suggest nuclear power sources. In this sense, the mass-energy .■elation of relativity is important to space travel. But other pre- dictions of relativity, such as time dilatation, are probably not significant to space travel. If we could burn nuclear fuel so efficiently that one-tenth of the initial mass of the spaceship were converted into kinetic energy, the final speed would be less than 0.5c. This would give a very small (0.14) time dilatation — hardly enough to allow one generation of voyagers to reach destinations outside the solar system. PROBLEMS 1. An atom moving at a speed of 1.0 X 10" m sec ejects an electron in the forward direction with a relative speed of 2.0 X 10" m/scc. Find 202 Looking In: Atomic and Nuclear Physics the electron's speed as seen by an observer at rest (a) using a Newtonian transformation and (6) using a I.orentz transformation. Arts, (o) 3.0 X 10* m/sec, (6) 2.7 X 10 s m/sec 2, Find the length of a meter stick when it is moving at a speed 0.90c relative to the observer. Consider the cases when the stick is oriented («) parallel and (6) perpendicular to its direction of motion. Arts. («) 18.5 em, (b) 10(1 cm 3. What speed will an electron have to acquire for its relativity mass to be twice its resi mass? Ans. 2.5 X 10 K m/sec 1. What is (he energy equivalent of the mass of an electron? Ans. 0,51 Mot 5. What is the radius of curvature of the path of an electron whose kinetic energy is 20 Mev when moving perpendicular to a magnetic induction of 0. 10 weber/rn-? Attn. 0.6S m h. Imagine that you are moving with a speed |c past a man who picks up a watch and then sets it down. If you observe thai be held the wmIcIi for (>,0 sec, how long does he think be held it? {Hint: You want It — U when you know t' t — l\.) Ans. 4.0 sec 7. From the mass-energy relation, calculate the energy released in the reaction ,H= + ,H* - tHe*. (Data: ,H« = 2.014743 amu, Mr* = 4.003874 amu) Ans. 24 Mev 8. A meson has a lifetime / = 1.0 X 10~*see before it decays. Find how far a meson with t> = 0.09c can travel. Ans. 300 m 9. Find the energy liberated when an electron and a positron annihilate. Ans. 1.02 Mev 10. If one uses the nonrehitivistic formula E,. = },m,r~. does one overestimate nr underestimate the kinetic energy of a particle of rest mass »io and speed vl 15 Hydrogen Atom Bohr Model ... for the value of his study of the struc- ture of atoms and of the radiation emanat- ing from them. Nobel Prize citation for Niels Bohr, 1922 By 1011, two rival pictures of the structure of an atom had evolved. J. J. Thomson suggested a "currant pudding" model of the atom iu which the positive charge was spread throughout a spherical volume of radius about 10 -ll> m, with electrons vibrating about fixed points within this sphere. Ernest Rutherford sug- gested a nuclear model of the atom in which the positive charge and almost the whole mass were concentrated in a very tiny nucleus; the electrons roamed through the rest of the atom, out to a radius of about 10~ 10 m. In crucial experiments, II. Geiger and E. Marsden probed the atoms in thin metallic foils with fast (" — sV c) a particles and showed that the observed deflections could be explained by the intense electric field near the center of a nuclear atom. Building on Rutherford's nuclear picture and using Planck's quantum hypothesis, X. Bohr fashioned a model of the hydrogen atom which explained its characteristic line 203 204 Looking In: Atomic and Nuclear Physics spectrum and correlated this with electrical measurements of excitation potentials and the ionization energy. 15.1 NUCLEAR ATOM REVEALED BY ALPHA SCATTERING a particles are helium ions (He ++ ) and are emitted spontaneously by some radioactive substances. In the Geiger and Marsden apparatus (Fig. 15.1), a particles are directed against a thin Fig. 15.1 Apparatus for investigating rt particle scattering, showing: radioactive substance fi, the source of a particles, thin foil F of scattering material, zinc sulfide screen S, and microscope M. a particles emerge from a channel cut in the lead block I, strike foil F, and ore scattered to screen S. The conical bearing allows rotation of microscope and screen about ver- tical axis FF. [H. Geiger and E. Marsden, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25. 604 (I913).l metallic foil F in an evacuated chamber. The number of a par- ticles scattered at various angles with the original beam direction is found to decrease with increasing angle, but some a particles are scattered at angles greater than 90°, up to 180°. Rutherford found this "almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you." For the IIe ++ ion is roughly 7,:i00 times the mass of an electron, and therefore the large deflections of a particles cannot occur by single collisions with electrons. Also, the foil used is so thin that a large Hydrogen Atom— Bohr Model 205 a deflection cannot result from several successive collisions with electrons. But, Rutherford reasoned, if all the positive charge and most of the mass of an atom are concentrated in a very small nucleus, then the a particle can come very close to a large amount of charge all at once, and it will experience a large deflecting force. Further, since the mass of the deflecting nucleus is greater than that of the a particle, backscattering is possible. Rutherford derived an equation for a scattering based on the assumptions that the nucleus and a particle behave as point Fig. 15.2 particles atoms. Deflection of a by nuclear-made! W^ Incident a particles Target positive charges, that Coulomb's law applies to the mutual repul- sion even at small distances, and that ordinary Newtonian mechanical principles hold (conservation of energy and conserva- tion of momentum). The number of a particles N reaching unit area of screen at distance r from the scattering foil was predicted to depend on r 1 4VW AV »i» 4 <»/2) (15.1) where A' a = initial kinetic energy of a particle N ( = number of a's incident per unit time on foil of thick- ness / having n target nuclei per unit volume Z = nuclear charge 2e = a's charge 206 Looking In: Atomic and Nuclear Physics In their precise and very readable report, Geigcr and Marsden neatly tabulated the results of counting thousands of a particles to show that .V was found to be proportional to (a) the thick- ness / of the scattering foil, (M the square of the nuclear charge Ze (using foils of Au, Ag, Cu, etc.), (c) the reciprocal of sin 4 (0/2), where B is the angle of deflection, and ((f) the reciprocal of the square of the initial energy K a of the a particles (using different radioactive sources). The Geiger and Marsden experiments verified Rutherford's nuclear model of the structure of an atom. They clarified the meaning of the atomic number Z and showed it to be more sig- nificant than atomic mass in ordering elements in relation to chemical properties. An upper limit of 10 IS m was obtained for the size of the nucleus, in terms of distance of closest approach of a particles. The validity of Coulomb's law was verified down to about this distance of separation between charges. When Geigcr and Marsden used still more energetic a particles in their deflection experiments, some deviations from the scatter- ing pattern predicted by Kq. (15.1) were observed. This was the first hint of the existence of a "nuclear force" of attraction in addition to gravitational force and the electrostatic (Coulomb) force of repulsion. 15.2 DATA FROM SPECTROSCOPY A grating spectrograph (Sec. 9.12) disperses the light incident on its entrance slit and focuses on a photographic plate a line image of the slit for each different wavelength present. As fine diffrac- tion gratings became available, owing largely to the skill of H. A. Rowland (1848 1901), spectroscopists diligently accumu- lated a vast number of measurements on the radiation emitted by atoms when excited in electrical discharge tubes, in ares, and otherwise. In general they found that (I) each element has its own characteristic line spectrum of wavelengths A or frequency v, (2) spectrum lines are generally sharp; elements producing the sharpest Hues are very stable; ('.',) lines in a sped rum may differ in relative intensity and in degree of polarization; (4) to the spectrum of every element can be ascribed a series of "term values" such that the frequency of every observed spectrum line can be obtained by differences of these term values. Hydrogen Atom— Bohr Model 207 We shall consider the much-studied spectrum of hydrogen, the simplest atom. Its spectrum comprises several well-defined groups of lines: the Lyman series in the ultraviolet, the Maimer series in the visible region, the I'aschen series in the infrared, and others still farther oul in the infrared, Fig. 15.3. Balmer limit Visible p asc hen limit Lymon limit \ "■ *- j . '•« 1 1 i r i i .1 -i_ j 4,000 8,000 1 2,000 16,000 20,000 Fig. 15.3 Some series of lines in the spectrum of hydrogen (wavelengths in Angstrom units). As a first step in developing au acceptable theory of atomic spectra, Rydberg found a relation which allows one to calculate the wavelengths in the hydrogen spectrum from differences between terms: l = R w ~ }y Khvve R = Lm x 10 ~ 3 A ~' < I5 ' 2 > When rtf = I and «,■ is given successive values, 2, 3, 4, 5, . . . , the differences of the terms in the Rydberg equation give the wave numbers 1 ,/X for the lines of the Lyman series. When n f = 2 and it; = :>. !, ."», .... the Rydberg equation gives the wave numbers for lines of the Halmer series, etc. Although this formula was obtained empirically, it turns out to be closely related with the way the spectrum originates. 15.3 BOHR'S THEORY There is a similarity of the hydrogen atom and our planetary system, in that in each case there is an attractive force inversely proportional to the square of the distance between the bodies. Bohr accepted Rutherford's concept of the nuclear atom and devised a model of the hydrogen atom in which orbital motion of the election was used to predict wavelengths of radiation which agree very closely with the observed wavelengths of the spec- trum lines (Table 15. 1), 208 Looking In: Atomic and Nuclear Physics Table 15.1 Some term values end energy levels for hydrogen Wovenos. (1/cm) Joules * Elec. volts ty H H t> O ■~ N ■nT cm r*> •o fo CN CM O <0 S PI in Batmer series u ti ■j u ^ 00 4 i CM o 8 CM 00 & d 3 s 3,047 4,387 6,855 12,184 27,419 -6.0x10" 20 -0.38 -8.7 -13.6 -24.2 -54.3 -0.54 -0.85 -1.51 -3.39 109,677 -217.3 ■13.58 man series 6562A 4861 4341 4102 Sol tier series 15, 233c 20,264c 23,032c 24,373c 3646 (Limit) 27,419c The following assumptions are made in the Bohr theory of the hydrogen atom: (1) The electron moves around a stationary nucleus (a good approximation, since m, iue = 1,830m,.). (2) The electron is held in a stable circular orbit by the Coulomb attrac- tion between the negative electron and the positive nucleus. (3) Only certain (quantised) orbits are possible for the electron, namely, those for which its angular momentum is a whole-number multiple of h/2-ir, where h is Planck's constant. (4) Radiation is emitted (or absorbed) by the hydrogen atom only when the elec- tron undergoes an energy change in a transition from one orbit to another. The energy of the photon emitted (or absorbed) is given by hv - K, - /:, (15.3) Newton's laws of motion are assumed to lie applicable to the hydrogen atom, just as to bodies of larger dimensions. The force Hydrogen Atom Bohr Model 209 of attraction exerted by the nucleus on the electron has the magnitude F = (l-,,.w- (15.4) The electron, moving with uniform circular motion, experiences a central acceleration a e = r 2 /r and a centripetal force mr- r, from Newton's second law. We equate the Coulomb force and the centripetal force r- (4xe u )r 2 Hi !.' - (15.5) The kinetic energy of the electron is A* = £mr*. If, convention- ally, we take K p — when the electron is far from the nucleus, the potential energy /*.'„ of the electron in orbit is E = p (4irt«)r so its total energy is E = A* + K„ - Smv ! - (Smb> from which Eq. (15.5) gives I s" e» /; = 2 (4jre«)r (-lvtn)r 1 c 2 2 (4Te )r (15.6) (15.7) (15.H) Of course, the kinetic energy of the electron is positive, but its total energy in a stable orbit is negative since it is bound to the nucleus, and work equal to |/i'| must be supplied to remove the electron from the atom (process of ionization), Fig. 15.4. Bohr's third assumption says that the permitted values of electron angular momentum are mm- = n rr n = ' i % 3, 4/W (15.9) The radii of permitted orbits are obtained by solving Kqs. (15.5) and (15.9) for the quantity (mr) 2 : (mv)i - jEsp and (m » )s 210 Looking In : Atomic and Nuclear Physics and equating the results to got - n' n _ 1, 2, 3, . . . 4jr*me ! -l^me 4 From Eq. (15.8), the total energy can he written (15.10) Sn = - I 2 (4we D )r (4jreo)W n = 1, 2, 3, . . . (15.11) These are the only energy levels possible for the hydrogen atom in the Bohr theory. The energy values for levels I to arc indicated t Totcl energy E •:• UnquanHzed O Radius r — — __ f Tj i ""■""-■■s^Binding energy r 3 y l\' / ^^. , Ionization P= - — -*\ MM srgy 1 + Fig. 15,4 Bohr's model of the hydrogen otom. in Table 15.1. The frequencies of radiation which the atom can emit or absorb are predicted from Eqs. (15.3) and (15.1 1) as A',- - E, 2tt'-W / 1 1 \ (15.12) Hydrogen Atom— Bohr Model 211 Mini 1 = " _ 2 * im < A (A L\ X c (4rco)%^e \n/ n, s / (15.13) li = = 1.097 X 10 s A-> from which the value of the Rydberg constant R can be verified as 2xW (■iirtn)''/r'r The orbit for which n = 1 is referred to as the lowest state, the ground state, or the normal state for the hydrogen atom. If an electron in the lowest energy state receives 12.07 ev of energy by collision with an electron or by absorption of a photon, it can be "kicked up" into energy level n = 3 (see Table 15.1). The time interval before the electron spontaneously drops back to a lower energy level is called the lifetime of the excited energy state, and is ordinarily about 10 -s sec. The electron we. are considering might drop Brat from state n = 3 to « ■ 2, then from «. = 2 to n - 1. It would thus be responsible for the emission of two photons. One would have the frequency of the first (H„) line in the Balnier series; the other would contribute to the first line in the Lyman series. The ionization potential is defined as the energy needed to remove, from an atom an electron initially in the lowest energy state. The lesser energy needed to promote an electron from one state to another of greater energy is called an excitation potential. Obviously the hydrogen atom lias only one ionization potential, but several excitation potentials. An atom with many electrons lias a corresponding number of ionization potentials. Because of the Pauli exclusion principle (Chap. 10), only in II and He do all the electrons have it = I in the ground state. In other atoms, the ground state is taken as the state of lowest energy. 15.4 EXTENSION TO HYDROGENLIKE IONS Bohr's model and theory apply successfully to ions which have only one electron, that is, I!e + , Li ++ , Be s+ , etc. The equation for the Coulomb force is modified to read F = e(Ze) / (■iwuijr* where Ze is the charge on the nucleus. This leads to inclusion of Z s in Eq. (15.13), and thus I-W-L -JL\ (15.14) X \nr my 212 Looking In: Atomic and Nuclear Physios Hydrogen Atom— Bohr Model 213 This relation predicts that He + (Z = 2) should radiate a series of iiiies hi the visible region for transitions to «/ = 4 similar to the Balmer series for H (Fig. 15.5). This Pickering series for Hc + was He* *t ■4 - ■3 - — 8: — 6- Fig. 15.5 Comparison of energy levels for H and for He 1 . n = \ n = 2 observed first in star spectra and was subsequently identified witii a laboratory helium light source. 15.5 CORRECTION FOR CENTER-OF-MASS ROTATION The frequencies in the Pickering series for He + are not precisely the same as those in the Maimer series for H, as Bq. (15.14) pre- dicts. Also, the heavy isotope II* has spectrum lines slightly shifted in frequency from those which Kq. (la. t:i) predicts should M CM Fig. 15,6 Rotation about center of mass (CM). be identical for both H 1 and TI-. These discrepancies suggest thai instead of simply considering that llic electron moves around a fixed nucleus, we should consider that both electron and nucleus move about their common center of mass (Fig. 15.6). Let r f and r„ be the distances from the center of mass to the electron and to the nucleus, respectively. Then r = r e + r„ is the distance. between electron and nucleus. If we introduce the angular velocity w - »/r, [Kq. (15.5)1, for the centripetal force mi- (4r tu )r : becomes nr \ « = mr *>° (18.15) t-Mreojr- The equation quantizing angular momentum, Eq. (15.9), becomes mr,*a> + .l/r„ 2 u = n 7 r- 2ir (15.16) where .1/ is the mass of the nucleus. From the definition of center of mass, M m + ,1/ and r„ = m m + M By combining Eqs. (15.15) and (15.17), we find (4^jH = m ">* rwi and !•'*,.< = w m irw ■"fts- 2.TT where the symbol »i r ..,i is used for the "reduced mass," m\! m Wr,.i = ; m + .1/ 1 + tn/M (15.17) (15.18) ( 1 5. fit) < 1 5. JO. Since Fqs. (15.18) and (15.19) differ from Eqs, (15.15) and (15.16) for no nuclear motion only in the replacement of electron mass m by reduced mass »w, we see that the energy /■.'. corrected for motion of the nucleus is related to the uncorrected energy A'„ of Eq. (15.11) by B. = m "" /■:„ - m E„ 1 + w M (»-«)* (15.21) since m/M « 1 . If we use the corrected expression for energy in Eq. (15.11), the Bohr equation (15.13) for wave numbers becomes 1 u _2irV_ mM f \_ I \ X (47r e „} a A 3 c m + M \rf nf) (15.22) 214 Looking in: Atomic and Nuclear Physics This correction shifts each energy level by ahout 0.055 per cent for H 1 . For the isotope II 1 , the shift is less. Hence a frequency difference can be observed when two isotopes of an element are present in a light source. The first (II a ) line hi the Balmer series of II 1 has wavelength (in(>2.80 A; that for II 2 lias wavelength 6561.01 A. The reduced mass correction also explains why the energy difference, and hence the frequency of radiation, is slightly greater for the helium ion He + (say, from n = 6 to n = 4) than for the corresponding transitions (n = 3 to n = 2) for hydrogen. 15.6 THE CORRESPONDENCE PRINCIPLE Bohr's correspondence principle is the guiding idea that, in the limit, the laws of quantum theory must join and agree with classical theory (which does not involve It). This asymptotic approach is to be expected when we go from microscopic systems to those of larger dimensions, or for large values of the quantum number n. Bohr's theory for the hydrogen atom does show such agree- ment. With the aid of Kqs. (15.5) and (15. 10), we may express the frequency of rotation of an electron in a Bohr orbit as f'orb — tj-~ — v 2Hr I C4 ire u )mr ■]' = me* WllW (15.23) On classical (Maxwell) theory we should expect this electron to radiate energy of this frequency, and possibly its harmonics. But the theory which includes Bohr's quantum assumptions for the 11 atom gives for the frequency radiated me* (I _1_\ Now ii/ 2 nr -nf «,' «,-%/ (n, 4- n,)(nj - n f ) If rii and n; are both large compared to 1 and if An is small, we can write this approximation nf 1 n, s 2n Am 2An n 8 Hydrogen Atom — Bohr Model 215 where An = tii ~ n/ and n»?ij« %. Then the Bohr frequency becomes Vln.lir ~ me* 4« 2 A s a 3 An (15.24) Comparison of Eqs. (15.24) and (15.23) shows that for large orbits (large n) and for Are = 1 , the atom radiates the frequency expected from classical electromagnetic theory. For An > 1, we get harmonics. This is an example of a transition region between macroscopic and microscopic physics where the laws of classical physics and quantum physics overlap. QUESTIONS AND PROBLEMS 1. Which of the experimental observations mentioned in Sec. 15.2 are satisfactorily explained by the Bohr theory of the H atom? Are any not explained? 2. How can the fact that the spectrum lines of hydrogen are sharp be used to support the statement that all electrons have identically the same charge ef 3. At what temperature will the mean kinetic energy of hydrogen atoms be just sufficient to excite the H a line? Am. 93,40G°K 4. Selig Hecht showed experimentally that a dark-adapted human eye experiences the sensation of light when the retina is irradiated by as little as 10 X 10~ 12 erg. What is the minimum number of quanta of yellow light (5,893 A) which the eye can detect? Arts, about 3 5. Assume that a free electron having kinetic energy 24.2 X I0 -so joule unites with a H + ion, goes to the lowest (n = 1) level, and gives up its energy in a single photon. What is the frequency of the photon radiated? Ans. 36 X 10'Vsec 6. How much energy is there in a quantum of violet light, wave- length 4,358 A? In a quantum of yellow light, wavelength 5,893 A? Ans. 2.84 ev, 2.10 ev 16 Quantum Dynamics In this paper 1 am going to attempt to find the foundation for a mechanics of quantum theory. This mechanics is based exclusively on relations be- tween quantities which are observable in principle (e.g., frequencies and in- tensities of line spectra, and not elec- tron orbits), ... W, Heisenberg, 1925 Beginning with Bohr's initial formulation of the quantum theory of atomic structure in 1914. physicists recognized that the mechanics of systems of atomic dimensions must obey laws differ- ent- from the larger systems successfully described by the classical mechanics of Newton. By 11)24, a new method of treating atomic phenomena began to be developed. It is known as quantum mechanics, quantum dynamics, or wave mechanics. The names of L. de Broglie, K. Sell nidi tiger. W. Heisenberg, P. A. M. Dime, ami EL U. Condon arc chiefly associated with this development. The concepts discussed in this chapter bring us to an acecpt- 216 Quantum Dynamics 217 able theory of atomic physics. We arrive at a logical branching point in our path. Armed with a successfully tested theory of the atom, we can now (I) try to understand and predict properties of atoms in intimate aggregation (solid-state physics) or (2) we can turn to investigation of the internal structure of atoms. A goal of such nuclear studies might be ultimately to manipulate nuclear particles to our use, as a chemist manipulates atoms to create molecules with desired proper! tee. 16.1 PARTICLES AND WAVES Planck's derivation of the law for the energy distribution of blackbody radiation (1900) first brought to light the particle (quantum) aspect of electromagnetic radiation. Einstein strik- ingly established this viewpoint with his explanation of the photo- electric emission of electrons from solids (1905). Photons were endowed with momentum {hv/c) by the Conipton effect (1924). Also in 1924, Louis de Broglie, proceeding from relativity theory and the observation that nature is symmetrical in many ways, suggested that whenever there are particles witli momentum p, their motion is associated with (or "guided by") a wave of wavelength V (16.1) The square of the amplitude of the de Broglie (matlcr) wave in a given region is interpreted as being proportional to the prob- ability of finding the particle of momentum p in that region. In de Broglie's hypothesis about wave-particle duality, an electro- magnetic wave tS the de Broglie wave for a photon, and proceeds with speed c. The de Broglie waves for electrons, protons, neu- trons, etc., are That electromagnetic waves, but "matter waves," which travel with the speed of the particle. We shall now discuss (I) a verification of these de Broglie waves and (2) something about how their value at various points in space may lie calculated. Since the de Broglie equation predicted that 100-ev electrons should have wavelengths of about I A, it was suggested that the wave nature of mailer might be tested in the same way that the wave nature of x rays was first tested. A beam of electrons of 218 Looking In: Atomic and Nuclear Physics appropriate energy could be directed onto a crystalline solid (Fig. Hi. In). The atoms of the crystal form a three-dimensional array of diffracting ©enters for the de Broglie wave guiding the electrons. There should be strong diffraction of electrons in certain directions just as for the Bragg diffraction of x rays. (a) O (c) Fig. 16.1 (o) Davis son and Germer apparatus, {fa) Angular distribution of secondary electrons, (c) Interpretation in terms of Bragg reflection of electrons (refraction of rays has been omitted). This idea was tested by C. J. Davisson and L II. Germer using 54-ev electrons and a crystal of nickel (Kig. HUfe). The emerge] i ( beam showed an intensify peak for 6 = 50°. The wave- length calculated From the Bragg equation turns out to be just h/p for a 54-ev electron. (The fact that electrons are observed at other angles is attributed to secondary emission: Some incident Quantum Dynamics 219 electrons collide, with and share their kinetic energy with some of the electrons in the solid, with the result that some of these are emitted at random angles.) Experiments on electron diffraction confirms the hypothesis that their motion is directed by a wave of some kind, and the wavelength agrees with that predicted by the de Broglie relation, A = h/p. 16.2 DIFFRACTION OF PHOTONS AND NEUTRONS De Bro^lie's hypothesis suggests that particles of any type may exhibit diffraction effects. The diffraction of neutrons has been useful in the investigation of the structure of solids. Beams of neutrons whose wavelength is roughly equal to the spacing of atoms in a solid can be obtained from a nuclear reactor. These beams are diffracted by layers of atomic nuclei. On the other hand, x rays are diffracted from planes in the solid where the density of electrons is highest. Thus the two types of experiment can give supplementary information about the structure of a solid. X-ray investigations reveal the location of the (bound) electrons in a solid ; neutron diffraction reveals the arrangement of the nuclei. 16.3 WAVE MECHANICS We expect that a de Broglie wave will obey the same type of second-order differentia] equation (Appendix C) used to represent other waves (Chaps. 9 and 12). Important applications of Schrodiugor's equation are to cases where the electron is subject to forces which hold it in a certain region, as in an atom or in the atomic lattice of a metal. The potential energy of the electron then varies from point to point. As the simplest case of this type, let us examine the wave function i>(x,i) for a particle of mass m which can move along a line between stops a distance L apart, like a bead on a stretched wire. Fig, 16.2 Particle confined to linear motion within range L. m -o 220 Looking In: Atomic and Nuclear Physics The particle will never be outside the interval < x < L; so ^ is zero for x < mid x > L. Inside the region considered f here are no forces on the particle; it is a free, particle. So the wave equation has sine and cosine, solutions, hut these must he zero at the. ends of the allowed interval. The allowed wavelengths of the de Broglie wave arc A = 2L/n, which leads to h nh P = \ = 2L (lfi.2) showing that the linear momentum is quantized. The kinetic energy of the particle is p a I ii' ; h- k ~ 2m~ 2m 4JS 8mP (18.3) and since we have taken K v = 0, the total energy must have one of flic values The particle is located by the matter wave iff = ( ^D sin - . J cos ait n = 1 , 2, 3, , . . (16-4) (16.5) The amplitudes of the standing waves for states of motion cor- responding to n = 1, 2, 3, ... , vary as shown in fig. 1(5.3. (There is a close analogy with standing waves in a vibrating string.) We see that the act of localizing or hounding a particle leads to the requirements that (1) the energy of the system can take on only certain values and (2) zero is not a possible value of the kinetic energy. Another important type of prediction from wave mechanics deals with the "leakage" of particles across an energy barrier. Suppose we have a particle bound in a shallow potential energy "hole" (1'ig. Hi. 4). There are now two kinds of solutions for the wave equation. There are solutions for any K > 0. Particles in these states have enough energy to escape; ^ extends over all space for them. Hut for particles whose lit is less than #„n (Tig. 1G.4), the total energy is negative, and for K < the wave equa- Quantum Dynamics 221 tion has solutions for only certain values of the energy. The higher the energy the more nodes there are in the wave (Fig. 16.46). The solutions are sinusoidal inside the well and have exponential tails outside. Thus (here is some probability for lind- Fig. 16,3 Wave functions for a bead on a string, for states n = 1,2,3,4. '1 4 x— *- '. t >■■ '>i (a) X-— \L Fig. 16.4 (a) A "square" potential hole, and (b) the wave function of its states n = 1,2,3. ing particles in a region where, according to classical theory, they do not have enough energy to be. Around a nucleus we may think of a potential harrier whose craterlike shape is determined by the Coulomb electrostatic force and a shorter-range 1 force of nuclear 222 Looking In: Atomic and Nucfear Physics attraction. The wave viewpoint predicts that, charged particles which do not have enough energy to go over the top of this barrier have a small but not zero probability of occasionally tunneling through the barrier. 16.4 BOHR ORBITS OR DE BROGLIE WAVES? If we apply the concept of matter waves and the probability interpretation of Sehrodinger's equation to the hydrogen atom, we find that I he features which the Bohr theory correctly pre- dicted (only with the aid of arbitrary assumptions: mr = nh/'lir, etc.) follow as a natural outcome of the mathematics involved. The quantum dynamical treatment provides additional informa- tion as well. The electron in a hydrogen atom has potential energy -(■'■ \irt„r. If we write for the radial distance r - V-t 2 + y* + z 1 Sehrodinger's relation |Appendix C, Kq. (4}| becomes (16.fi) The solution of this equation, ^(r,ij,z), is a function which has a definite value at each point in the neighborhood of the nucleus. To discuss this equation, it is convenient to change from reetan- Fig. 16.5 Rectangular and spherical co- ordinates. Quantum Dynamics 223 gular coordinates to spherical polar coordinates, using the relations r = distance of point a from origin = y/x 2 + y* + z z 8 = angle from z axis to r = cos~ l {z/r) 4> = angle around z axis measured from x axis = tan -1 (</ as) With the introduction of the coordinates r, Q, $, Kq. (Ifi.O) can be separated into three ordinary differential equations, a fact we represent by f(r,e,<t>) = rt(r)0(0)*W (lfi.7) The function li describes how f varies as we go out from the nucleus in a definite direction. The functions 8 and <$ describe how ^ behaves from point to point on a sphere of radius r. The equation for the function K(r) has a solution for any posi- tive value of E. These solutions correspond to states in which the electron has enough energy to escape from the atom; there are no quantum restrictions on the energy of a free electron. But there are only certain negative values of E for which Kq. (16.7) has any continuous solution. When the electron is bound in the atom, an acceptable wave function ^ exists only if E has one of the particular values E„ = -me 4 1 -I3.fi ev n = 1, 2, 3, (10.8) These are the same values for the energy states that the Bohr theory predicted. The quantum number n is here related to the part R(r) of the wave function which describes the probability per unit volume of finding the electron in a given volume element at various distances from the nucleus. This is independent of 8 and *. We can compute the average distance of the electron from the nucleus by averaging over the probability distribution. The result is roughly the same as the radius of the first Bohr orbit. The energy (Kq. 10.8) is in exact agreement with the Bohr theory. For each value of n, the equation for 8(8) is found to have one or more solutions, described by a second quantum number I. This quantum number takes on only the values / = 0, 1,2,3, n - 1 (16.9) 224 Looking In: Atomic and Nuclear Physics Solutions of the * equation arc related to solutions of the R equa- tion such that the electron is less likely to he found near the nucleus when in a high-/ state than when in a low-/ state of the same energy. For each value of /, the equation for *(<£) is found to have one or more solutions, designated hy a third quantum number jtcj. This takes on only the values m , = -/,_(/- i), -(I -2), — 10 12 (I - I), I (16.10) Xo solutions of Schrodinger's equation for the hydrogen atom exist for any other values of «, /, and m ( . 16.5 THE QUANTUM-NUMBERS GAME An atom can be completely described by the use of just four quantum numbers for each electron. Three of these we have already introduced. The principal quantum number » determines the energy, E<|. (10.8). It may have the integral values n = 1, 2, :i, . . . . The orbital angular- momentum quantum number I deter- mines the angular momentum of the motion of the electron about the nucleus. It may take on any integral value from to n — 1. The corresponding value of the electron angular momentum i- VKI+ 1) h/2n. The component of the orbital angular momentum along the a axis is given by null, 2x, where m may lake on any of the 2/ -j- I values: 0, ±1, ±2, . . . , ±1. The quantum number mt is called the magnetic quantum number because physically the presence, of an external magnetic field is necessary to establish a reference direction (z axis) in space. In a magnetic field, the electron's angular momentum is said to be "space-quantized" because its component along the di recti >i the magnetic field is restricted to the values mji 2tt (Fig. l(i.(i). We now introduce a fourth quantum number 8, the electron- spin angular-momentum quantum number. This quantum num- ber defines the internal angular momentum {and associated mag- netic moment) which an electron is found to have, independent of its orbital motion. An experiment to show this property of the electron was performed by Stern and Clerlach. If a neutral atom Quantum Dynamics 225 which has a magnetic moment passes through a uniform magnetic field, it experiences a torque, but no deflecting force. If, however, the field is nonuniform, the atom experiences a net deflecting force as well. Consider a beam of II atoms. The electron in the normal state has zero orbital angular momentum for n = 1, / = 0, mi = 0. There is no magnetic moment due to orbital motion. Vet the beam of II atoms is observed to split into two parts, each associated with a restricted orientation of the clec- Fig. 16.6 Possible orientations of angular-momentum vectors. tron's spin angular momentum. The two possible values of the component of the spin angular momentum in the direction of the magnetic field are ±%k/2*. We conclude that, unlike the other quantum numbers, which are integers, n can have only the value £, The component of the spin angular momentum may be either parallel or antiparallel with the applied magnetic held. So we can define a spin magnetic quantum number m„ - ±\ and write the component of the spin angular momentum in the direction of the applied field as mji/2w. («> J Traces on receiving I Direction atoms move 1 Fig. 16.7 Magnetic field used in Stern-Gerloch experi- ment, (a) With no field there is a single beam, (b) With field, beam splits; some atoms are deflected toward N po!e, some toward S pole. Traces where beam strikes detecting plate are shown at top. (Adopted from R. 0. Rusk, "Afomk and Nuclear Physics," Applet on- Century -Crofts, Inc., New York, 1958.) Quantum Dynamics 227 16.6 THE PAULI EXCLUSION PRINCIPLE In 1925 W. Paul! suggested that a complete description of the atom must include a unique description of each electron in the atom. Xo two electrons in an atom may have identical values for a set of four quantum numbers. To see how this rule operates, consider the number of elec- trons permitted in the first orbital group (or shell) for which n = 1. Since n is 1, 1 = and m t = 0, But m, may be +1 or - '.. So in this first group there may be two electrons, distinguished only by having their spins in opposite directions. The continua- tion of this assigning of quantum numbers to electrons in many- electron atoms is shown in Table 16.1. Table 16.1 Numbers of electrons in groups (or shells) as determined by Pauli's exclusion principle Orbital group n / m s No. elec. in No. elec. in subgroup completed group 1 { 1 1 i 2s 1 2 Iii the terminology of Table 16.1, we replace the term "orbit" by "group" or "shell" (determined by n). This emphasizes the three-dimensional nature of the atom. The shells are often named the K, L, M, . . , ,Q shells, corresponding ton =» 1,2,3, ... , 7. 226 228 Looking In: Atomic and Nuclear Physics Within a shell, electrons with a common value of / form a sub- shell These arc designated s, p, rl, or / subshells according to whether / has the value 0, 1,2, or 3, 16.7 BUILDING THE PERIODIC TABLE OF ELEMENTS When the elements are arranged in order of increasing atomic number, a periodicity in their chemical properties becomes apparent, as shown by Mendeleev. The structure of the periodic table is in agreement with the ideas of filled shells and subshells as predicted by the 1'auli principle. We may "build up" an atom by putting each electron in the shell of lowest energy until the quota of permitted states is filled. Any additional elections must- be put. in the next shell as shown in Table lfi.2. The final column of Table KS.2 is a description of the electrons in the outside shell for the normal (ground) state of the atom. The electron configura- tion of an atom is described by the abbreviated not at inn ui" t he last column. For example, %- means there are two electrons in the n - .i, I = I subshell. The quantum numbers we are using were originated for the case of one electron. J I is remarkable thai by assi g nin g occupied states in terms of these numbers we get ati accurate description of many of the properties of complex atoms. Kvidently the various Table 16.2 Electron configured ion for 1 ght atoms >»,/-* Z 1,0 2,0 2,1 3,0 3,1 3,2 4,0 Configu- Element (U) (2s) (2 P ) (3*) (3 P ) (3d, (4s) ration H 1 1 If He 2 2 w Li 3 2 I 2* Be 4 2 2 2s* B 5 2 2 1 2p L 6 2 2 2 2p= N / 2 2 3 2p* O 8 2 2 4 2p 4 F 9 2 2 5 2p» Ne 10 2 2 6 2p r> No 11 2 2 6 T 3s Quantum Dynamics 229 electrons in a complex atom must disturb each other's orbits very little. One sort of disturbance, called screening, should be men- tioned. An outer electron is in a weak electric field because inner electrons screen it from the positive charge of the nucleus. Hence states in which the electron has some probability of being found very near the nucleus will have lower energy (greater binding) than those states in which the electron tends to stay outside the screening inner electrons. Of the solutions of the Schrodinger equation for a given n, those with lower values of J will tend to / 2.1 16 12 B 4 - _i I 1 I i i i ■ !■■! I ... I 10 I i 1 14 IS 22 26 30 X Fig. 16.8 Variation of ionization energy (J in ev) with atomic number Z, sug- gesting greater stability of certain electron configurations. penetrate the cloud of screening electrons most. Hence, for atoms containing more than one electron, penetration causes the energy of an orbit to depend on I. as well as on ti. (In terms of the liohr picture, energy depends on the shape of the orbit as well as on its size.) Klectrical measurements which correlate well with electron configurations are shown in I'ig. Ili.8. The ionization energy is the work needed to remove the least tightly bound electron from an atom. The variation of ionization energy with atomic number 7. suggests that certain electron configurations have relatively great stability. The first is for helium, where the it = I shell has 230 Looking In: Atomic and Nuclear Physics its ii"" 1 '' 1 <>f two electrons. The sharp drop to the binding energy for lithium is attributed to the fact that the third electron must bi> ndded io l lie n - 2 shell and is therefore farther from the nucleus. Tor the elements after lithium, there is a trend toward increasing binding energy until another maximum is reached at neon, when the n = 2 shell is filled. Like He, Xe is an inert gas. This variation in binding energy is repeated several times in the periodic table, each time giving a maximum binding at an inert gas, followed by a minimum for the succeeding alkali metal. The size of atoms also oscillates from shell to shell, about a value approximately 1 A for the radius. In each shell, the alkali metal has the largest radius. 16.8 CHARACTERISTIC X-RAY SPECTRA When a target is bombarded with electrons of high energy (l>), x rays are produced which have a spectrum which is continuous up to the maximum frequency given by the relation fcjWx = I <■■ In addition, x-ray spectrum lines arc observed at frequencies which are characteristic of (determined by) the target material. Characteristic x-ray spectra can now be explained in terms of the shell structure of atoms. First, a vacancy must he created by the displacement of an inner electron from, say, the K or L shell. Since there are usually no near-lying vacant energy levels to which these electrons may be promoted, they must be removed altogether from the atom (ionization). This may he accomplished when atoms of the target are bombarded by electrons which have been accelerated through a potential difference of many thousand electron volts or by high-frequency photons. Transition of a near- lying electron then occurs to Jill the vacancy. If a vacancy in the K shell is filled by an electron from the h shell, an x-ray photon is radiated whose frequency depends on the difference in energy between the K and L shells. The vacancy left in the L shell is in turn tilled by an electron from a still higher energy state, with radiation of a photon of somewhat lower frequency, Fig. lfi.9. Since the energy of the electron in the K shell is chiefly determined by the nuclear charge Z, Moseley found he could use the K a lines of the elements to identify the atoms in the target of the x-ray tube. lie found a linear relation between the square root of the frequency and (Z — l), as would be expected from Quantum Dynamics 231 the Bohr formula with allowance for screening by the inner K electrons. From the relation = Rc(z - ]y-(L- L\ (16.11) Moseley was able to prove that early assignment of atomic numbers to cobalt and nickel was in error. The atomic mass of a natural mixture of the isotopes of Xi is 58.(i!) and for Co 58.94. They were first placed in the periodic table in the order of increas- ing atomic mass. Hut the x-ray lines showed that this order should be reversed, for Z Co = 27 and Zy, = 28. Moseley 's work K 0.2 0.4 0.6 0.8 1.0 Wavelength, A (a) E, E M (b) K« K » L a H \ <M a M N Fig. 16.9 Characteristic x-ray spectrum, (a) Molybdenum target with V (a) Simplified energy level diagram. 35 kv. (1913) gave the first accurate method for measuring atomic number, Z. The committee awarding the 1917 Xohel Prize to C. G. Barkla, for his work on characteristic x rays, stated that Moseley would have shared the award but for his death at Callipoli. 16.9 PHYSICS OF THE SOLID STATE Our theory, based on the nuclear atom model and quantum mechanics, tells us that under ordinary circumstances of tem- perature and pressure the nuclei of atoms will never get very close to one another. The combination of atoms should therefore be 232 Looking In: Atomic and Nuclear Physics explainable through the exchange or sharing of electrons. In terms of tfic measured masses and charges it should be possible to describe the formation of molecules and chemical reactions. One might also hope to describe crystal lattices and the mechan- ical, thermal, electric, and magnetic properties of solids. Prac- tically, the difficulty is the complexity of the computations. We shall examine some of the successes of quantum mechanics in explaining important electric and magnetic properties of solids. This comprises hut one segment of solid-state physics in which there is very active research. 16.10 CLASSICAL THEORY OF CONDUCTION IN METALS A theory proposed by Drude and Lorentz, soon after the dis- covery of the election, assumed, as have later theories, that some of the electrons are free to travel throughout the whole volume of a crystalline material. In a "good" metal, it was assumed thai there is about one free electron per atom and that the number of conduction electrons is independent of temperature. These elec- trons dart around in all directions ui) h the high speeds of thermal agitation. Hut if an electric held is applied, the "electron atmos- phere" experiences a relatively slow drift, superposed on the random thermal motions. The electron drift is the electric current. The transfer of any increase in the energy of random motion in any direction constitutes thermal conduction. To make quantita- tive predictions, it is necessary to make some assumptions about the distribution of electron speeds. Theories have differed in these assumptions. The classical theory assumed that the electron speeds followed the same distribution law as .Maxwell and Bollzmanii had used for molecular speeds in developing a successful kinetic theory of gases (Chap. 8). Among a large number N of electrons, the frac- tional number N,'N having speed r is given by AT vV \2fc27 (16.12) Tf we plot this expression against o, the area under the curve between r, and v-> equals the fraction of all the electrons whose speeds are between r, and r*. Since kinetic energy depends on the Quantum Dynamics 233 speed squared, the average kinetic energy depends on the average of the squares of the speeds. The square root of this average is called the root-niean-square (rms) speed. The distribution curve becomes flatter and the maximum shifts toward higher speeds as the temperature increases (Chap. 8). The classical theory gives rough predictions of the electrical and thermal conductivities of metals, if is in accord with the experimental observation that the best conductors of electricity are also the best conductors of heat. Wiedemann and Franz ( 1 850} showed that the electrical conductivity thermal conductivity ratio is a constant, for metals. The classical theory, using known values for e and k, predicts that the thermal conductivity/ electrical conductivity ratio = li.ll X 10 _B 'T cal ohm sec (°K). Fig. 16.10 Hall effeel D D' —n — This checks well with values measured for platinum and other I mre metals. But the classical theory meets with significant failures. It predicts that the free electrons should contribute |/f to the specific heat of a crystal. This considerable electronic specific heat- is not observed experimentally. Also, the theory is unable to explain the enormous range of electrical resistivity for different materials. Further, the theory suggests that since the free elec- trons have magnetic moments, even a weak magnetic field should produce a large paramagnetic magnetization (magnetic moment per unit volume) in a conductor. It does not. Finally, the theory has difficulty in predicting the sign of the Hall coefficient. For a current-carrying conductor (Fig. Hi. 10) one would expect that a potentiometer connected between Cand J), in a plane perpendicu- lar to the current . would indicate zero potential difference. If now an external field H is applied, the conduction electrons experience a magnetic thrust perpendicular both to H and their velocity v. The equipotential line CD is tilted through some angle <j> to position CD'. The classical theory predicts that tan 4> (the Hall coefficient) should have the same sign for all metals. It docs not. 234 Looking In: Atomic and Nuclear Physics 16.11 FREE-ELECTRON QUANTUM THEORY OF CONDUCTION Fermi introduced a radically different description of the free electrons iii a metal. He incorporated the exclusion principle, assuming that the "free" electrons hi a metal are quantized and that no two can act exactly alike. Momenta are quantized; only two electrons (having opposite spins} can have a given mo- mentum. As the temperature is lowered, electrons settle down hy quantised slops to lower momentum values. But as a consequence of the exclusion principle, some electrons wil! remain at mo- mentum values considerably above zero: thai is, linn- will have appreciable energy, even at absolute zero temperature. When the N f J0°K i 300° K \Very high temp. Fig. 16.11 Fermi distribution of speeds at various temperatures. temperature rises, only the electrons of highest momentum can accept thermal energy and move to still higher momentum values. The Fermi distribution law is expressed by N = h l e m, "' i!_A "-'* r 1 (16.13) where E,„ is the maximum energy an electron can have at 0°K. In Fig. 16.11, the progressive rounding of the curve as temperature increases represents the shift of some electrons to higher energies. The Fermi distribution curve should be compared with the Max- well distribution (Chap. 8). The ['ermi theory successfully accounts for the slight partici- pation of electrons in specific heats. In Fig. Hi. 12, the Fermi distribution of energy is plotted. At 0°K all energy states are Quantum Dynamics 235 occupied up to a certain maximum (Fig. 16.12a). At a higher temperature some electrons in upper levels have been able to accept energy ami move to still higher [ovate (Kg. [6.126). Btl1 owing to quantum restrictions, relatively few electrons have participated in the temperature rise. The Fermi theory predicts that electrons in a conductor should contribute roughly 1 per cent of the amount predicted by the Maxwell theory, in agreement with experiments in calorimetry. The fact that all energy levels, up to a certain maximum, are filled means that for every electron traveling to the right in a metal there is another elect ion traveling toward the left. Thus all electrical conduction in the metal must be due to the relatively few electrons near the top of the distribution (Fig. JO. 126) which Fig. 16.12 Fermi distribution of energies, showing (a) all levels filled up to a maximum of K, (b) some electrons promoted to higher energy levels at a high temperature. Rel, no Rel * no can be excited easily to an unoccupied quantum level. One con- cludes that electricity must be conducted by only a small fraction of the free electrons (rather than by all, as assumed in classical theory). In turn, this implies that an electron must be able in travel long distances without being bumped by ions in the crystal lattice. The free-electron quantum theory, like the classical theory, is unable to account for the distinction between con- ductors and insulators. 16.12 BAND THEORY OF CONDUCTORS, SEMICONDUCTORS, AND INSULATORS In the modem band theory of the electronic structure of solids, the effects of the lattice ions on the free electrons are considered to explain the occurrence of conductors, insulators, and semi- conductors. The moving electrons are pictured in terms of 236 Looking In: Atomic and Nuclear Physics do Broglie waves of wavelength \ - h/mr. The influence of the lattice ions arises from the variation of potential from atom to atom in the crystal (Fig. Hi. 13). The passage of I he do Broglie waves is treated mathematically by methods similar to those used in investigating the passage of light waves through a similar lattice. Yinnrvv Fig. 16,13 Variations of poten- tial along a one-dimensional crystal lattice. It turns out that the graph of electron kinetic energy versus momentum, instead of having the parabolic shape (Fig. Hi. 14a) which it would have in a conductor where there was no variation of potential, jumps discontiuuously for particular values of de Broglie wavelengths (Fig. 16.14/*). Not all electron momenta (a) (b) Fig. 16.14 Energy vs. momentum: (a) Assuming no variation of potential between atoms; (b) assuming a variation of potential similar to that of Fig. 16.13. are possible. From this point of view, the effect of the ion lattice is to preclude certain values of electron momentum and hence to leave forbidden energy gaps at these momentum values. The properties of conductors, insulators, and semiconductors ran now be interpreted in terms of the conduction bands (Pig, Hi. 15). If the highest energy band containing electrons is full and Quantum Dynamics 237 is appreciably separated from other bands (a), the materia! is an insulator. To produce a current in such a material, electrons have to be advanced across an energy gap large compared to thermal energy /,•'/'. In a conductor, however, the highest band containing electrons is not full (/*). Kven a small external electric field can Allowed (Empty) Allowed [Partly full Forbidden / £ w » « r Forbidden ; (Allowed i) (Full) (a) (6) Allowed (empty) -Forbidden E 9 , p 2skT Allowed (full) (C) Fig. 16.15 Distribution of electrons in bands in (a) an insulator, (b) a conductor, and (c) a semiconductor. produce an unbalanced momentum distribution {a current) by promoting electrons to energy states of small excitation. Semi- conductors arc an intermediate case in which the highest occupied hand is full (c), but. the energy jump to the next band is compar- able to kT. Increase in temperature would be expected to lower the resistance of a semiconductor. 17 Radioactivity The new discoveries made in physics in the last few years, and the ideas and potentialities suggested by them, have had an effect upon the workers in this subject akin to that produced in literature by the Renaissance. J, J, Thomson, in an address on radioactivity, 1909 Radioactivity has provided us with much of the knowledge we now have concerning the nucleus. Emission of a and ji particles by certain atoms suggested the idea that atoms are built of smaller units. Measurements of the scattering of a particles by atoms confirmed Rutherford's idea of the nuclear atom. The dis- covery of isotopes can be traced to the analysis of the chemical relationships among the various radioactive elements. The bom- bardment of atoms with energetic a particles from radioactive su Instances was found to cause disintegration of some atomic nuclei; this led in turn to the discovery of the neutron and to the present theory of the make-up of the nucleus. The transmuted atoms resulting from such bombardment are often radioactive. 238 Radioactivity 239 The decay of these artificial radioactive nuclides is in accord with the laws found earlier in the study of natural radioactivity. 17.1 TYPES OF RADIOACTIVITY In the theory of the nucleus there is no counterpart of the simple, easily visualized mechanical model employed in the Bohr theory of the atom. But the concept of energy levels, found so useful in studying the atom, is carried over to the description of the nucleus. Nuclear spectroscopy deals with the identification of these energy levels and is an important source of information about the nucleus, since radioactive changes can be measured with high precision. When the electronic structure of an atom acquires some extra energy, the atom almost always gets rid of this extra energy very quickly, returning to the ground state in roughly 10 - * sec. It does so by emitting one or more photons or an electron if there is enough extra energy. Many nuclei, however, can exist for long periods of time in an unstable state, that is, in a state from which the nucleus can and eventually will decay to a stable state. A nucleus may go to a state of lower energy by emitting an a particle (a radioactivity), an electron or positron (fi radio- activity), or a photon {7 radioactivity). Most "natural radioactivity" is found among the very heavy elements (A > 210), which tend to be unstable to a decay. These nuclei decay so slowly thai there are still some of them left from the time of formation of the elements. Radioactive isotopes not found in nature can lie prepared in nuclear reactions. 17.2 STATISTICAL LAW OF RADIOACTIVE DECAY The activity of a radioactive sample is defined as the number of disintegrations per second. The activity decreases with time. Each radioactive isotope has its own characteristic rate of de- crease, figure 17.1 is the plot of the decay of a radioisotope which decreases in activity by 50 per cent every 4.0 hr. The form of t In- experimental decay curve suggests that the decay is a loga- rithmic process. This is verified by plotting the logarithm of activity versus time. A straight line results. We can derive an exponential law of decay for a sample con- 240 Looking In: Atomic and Nuclear Physics tabling a large number of radioactive atoms. We assume that each undeeayed nucleus has a definite probability X of under- going decay in the next second and that this probability is inde- pendent of time and is independent of whatever other atoms are present. Then Iho number of decays in a time interval dt is equal to the number of undeeayed atoms present times the probability g < *m c 3 % ^ fc K T "> \ t„=^L. t \ i " 0.693 < 1 i i i .i 4 8 12 16 20 24 t, hr Fig, 17,1 Decrease in activity of a radioisotope with a 4.0-hr half-life. X (ll that each one of them will decay. Thus the change (decrease) in the number of undeeayed atoms is (IN = -\N(U (17.1) The decay constant X is the relative number dX/N of atoms which decay per second. The value of X depends only on which radioactive isotope we are considering, Uy separating variables in Kq. {17.1), we obtain a simple differential equation tlX N Xdt (17.2) whose solution is N = AV"*« (17.3) Radioactivity 241 where N a is the number of undeeayed atoms in the sample when t = 0. (Note the mathematical similarity with the equation for the exponential ahsorption of a beam of radiation.) The activity of a sample, the number of decays per second, is given by Activity = -.:- = XAV -W or Activity = XN (17.4) (17.5. The activity depends on the number of atoms present and on their decay constant, X. 17.3 HALF-LIFE The half-life T of a radioactive substance is the time interval in which the activity (and hence the number of undeeayed atoms) decreases by 50 per cent. For the activity of Fig. 17.1 this is 4.0 hr. Itoiii the definition that t - T when X = IN^ Eq. (17.3) becomes hN a = AV" W which gives T = log, 2 = 0.«»3 (17.6) (17.7) The average life '/'„, or life expectancy, of a radioactive nucleus may be calculated by Rimming the lives of all the nuclei and dividing by the total number of nuclei „, fi/ot dN I t - , A . . .. ., 1 ? « = / -57- = tt / 'Wr M dt = r Jo No No Jo X (17.8) The decay constant X is the reciprocal of the average life, in accord with the interpretation of X as the probability of decay of an atom per second. 17.4 UNITS OF RADIOACTIVITY A unit of activity was historically defined as the amount of radon (gas) in equilibrium with one gram of radium. The National 242 Looking In: Atomic and Nuclear Physics Research Council in 1948 extended this definition to define one curie as that quantity of any radioactive substance which gives 3.70 X 10 10 disintegrations per second. Since the curie is a rela- tively large unit, the millicurie (I mc = 0.001 curie) and the microcurie (I ^c = 10 -s curie) are widely used. A counter near a radioactive source detects a certain fraction of the particles emitted; the counting rate is proportional to the activity of the source. The specific activity of a radioactive source is the rate at which 1 gin emits charged particles. 17.5 GAMMA DECAY A nucleus in an excited statc(z*X i ) may go to a state of lower energy by emitting the difference in energy as a photon: zX* + hv C— M«v) (1 7.! I ) Now y decay does not cause a change in the atomic number or the mass number of the nucleus. The half-lives for y decay are seldom very long. Study of y radiation gives important information about the initial and final states of the nucleus undergoing a y transition. Like the spectra of atoms, the y spectra of nuclei are found to consist of sharp lines, showing that the nucleus has discrete energy levels. The observed energies of emitted photons give consistent results for the nuclear energy levels hv - Ei - E } (17.10) The electromagnetic-wave nature of y radiation is demon- strated experimentally by diffraction. This is feasible only for those 7 rays of relatively low energy because ruled gratings or crystals with effective spaeings about equal to very short y wavelengths are not available. The energies of high-energy 7 rays may be measured in several ways. When a 7 ray ejects a photoelectron from the inner shell of ati atom, hv = E k + I (17.11) where E k is the kinetic energy of the ejected electron and / is the binding energy of the shell from which it is removed. The ioniza- Radioactivity 243 tion energies (/) are known. Hence the energy of the y-ray photons may be determined by measuring the energies of the photoelectrons. Positron-electron pairs (Chap. 19) can be created by 7 rays with hv > 2m c'. The photon energy is transformed thus: hv = 2m,e* + E k + + E k ~ + E k . tceoil (17.12. From conservation of momentum, the recoil velocity of the nearby nucleus should be small. Its energy can generally be neglected. Measurements of the momenta of the electron and positron in a magnetic field then give information from which the energy of the X ray can he found. 17.6 ALPHA DECAY When an a particle is ejected from the nucleus, the original nucleus loses two protons and two neutrons. Its mass number decreases by four units while its atomic number Z decreases by two. « decay thus causes transmutation of the parent chemical element into a different chemical element Z X* -» «_,**-* + -.He' + Q (energy) -17.1:;. Now a decay occurs spontaneously, without any external forces, and it provides kinetic energy (#*,„) for the ejected n part ideas well as some kinetic energy {E k , d ) for the recoil ''daughter" nucleus. Hence a decay cannot occur unless the total rest mass decreases. The decrease in rest energy is equal to the kinetic energy released, called the disintegration energy Q: Q = E k ,,i + E k ,„ = (m„ - m lt - w Q )c ! (17.14) To predict whether a nucleus will undergo a decay, we may com- pare its rest mass with the sum of the masses of the product nuclei. Actually we can use the masses of atoms instead of those of the nuclei. The same number of electrons are associated with the initial and final nuclei, so the electron masses cancel in the calculation of Q. From Eq, (17.13), Q = (m* - mr — trine)*? (17.16) 244 Looking In: Atomic and Nuclear Physics Exampk, Find the Q value for the disintegration t t>N'l u * —* tlic* 4- S8 Ce I4l >. Prom tables of isotope mosses: ♦He 4 = 4.00387 » a Ce l4 ° = 139.01977 143.95364 toX'!'*' - [43.95550 Produeta - 143.95364 m = 0.00192 Q = mc* = 1.79 Mev Example. In a decay, what fraction of the disintegration energy appears as kinetic energy of (he a particle? Conservation of energy and conservation of momentum in a decay require Q = Et.4 + E kia = lmj>S + £fli a p a * m a v a = m&t From (he momentum equation, v d - (»ia/»ij)e„. Substituting this in the energy equal ion. ive have or Q " ^-'(S) + 1 AY~ = a 1 + m„/m d (17.16) (17.17) If .4 is the mass number of the parent nucleus, then m a /m a =* 4/(A — 4) and **„ 4-4 (17.1 7«) Thus for large .4, the a particle gets most, but not quite all, of the dis- integration energy. An interesting feature of a decay called the tunnel effect may be illustrated by data for a particular ease. One can perform an experiment similar to the Kutlierford-Ceiger-Marsden scattering experiment (Chap. 15) using a thin foil of 94 U i!8 to scatter the 7.68- Mev a particles from mIV 4 (also called Ra C). One finds that the Rutherford scattering law is obeyed. Kvtdently the a particles from IV 4 do not have sufficient energy to get over the Coulomb barrier; they are scattered away from the l"' :,s nucleus. This is suggested in Fig. 17.2, which shows the potential- Radioactivity 245 / u" 8 t * n, t V, 7.68 1 Decay 4.20 Me> 1 Mev » / r Fig. 17.2 Coulomb borrier: scattering of a high- energy particle and tunneling of a low-energy particle. Fig. 17.3 Wave mechanical description of tunnel effect. energy curve of an a particle near a U m nucleus and a IV 4 a particle being turned away by the potential barrier. Contrast this with the following fact: U 2,s itself is an a emitter, emitting a particles whose kinetic energy is only 4.20 Mev. We have a paradoxical situation : The lower-energy U S39 a particle can cross a barrier which the higher-energy I'o 414 a particles appear unable 246 Looking In: Atomic and Nuclear Physics to cross. An explanation on the basis of classical physics is impossible. The wave nature of the a particle must be taken into account. When we use wave mechanics to describe an a particle in the nucleus, we find that a little of the wave function will "leak" through the barrier so that there is a small probability that the particle may be found outside (Fig. 17.8). According to wave mechanics, if the a particle has enough energy to be outside, then there is some probability that it will be found there. This probability is very small for U 2M and accounts, roughly, for the U 23 " 1 half-life of 4,f> billion years. The tunnel effect works in either direction, so some of the IV" a particles used in the scattering experiment must have penetrated the nucleus, but the fraction which succeeded was negligible. The probability of tunneling depends strongly on the height and width of the potential barrier. 17.7 BETA DECAY The /3 particles emitted from a radioactive source are shown by deflection experiments to be high-energy electrons. There are good reasons to believe that, these electrons do not exist in the nucleus but are created by a rearrangement of the nucleus into a state of lower energy. Any excess of energy over thai required to provide one electron rest mass (m.c-) appeal's as kinetic energy of the emitted electron. An argument against the existence of electrons in a nucleus, prior to emission, makes use of the uncertainty principle. If an electron were confined in a region of dimensions no larger than about 2r = 1.4 X 10 _ " tn, the electron would have momenta as high as Ap = = 3.8 X 10 -' kg-m/sec = Um<c ■ttt &,v and hence kinetic energy as high as E* = \/{Ap')*c s + m<c* - m.c' 1 = 14m,c s = 7.2 Mev It seems unlikely that there are attractive forces in a nucleus which are sufficiently strong to bind an electron having this much energy. Radioactivity 247 Two different types of decay occur: ff~ decay, in which an electron is emitted from the nucleus, and (3 + decay, in which a positron is emitted. If the nucleus consists of neutrons and pro- tons only and if electric charge is conserved, then upon emission of an electron, a neutron must be converted to a proton, hZ = -f- 1. Similarly, positron emission involves the conversion of a proton to a neutron, AZ = — 1. (17.18) (17.19) For £~ decay to occur, the mass of the decaying nucleus must be greater than the mass of the product nucleus plus the mass of an electron. An atom which is heavier than the atom with Z one unit greater but with the same .1 will decay into that atom by 0- emission. The condition for £+ decay is slightly more complicated. Q — nix — wiy — '2m r c 3 (17.20) where m\ and m\- are the masses of the initial and final atoms, respectively, and hi. is the rest mass of an electron. An atom is ftf + unstable if it is more than two electron masses heavier than the atom with the same .1 and one less Z. There is still a third ($ decay process whose over-all result is the same as /J + decay. A nucleus may absorb one of its orbital electrons. This process is called A" capture since the elect Tons ill the nearest (re = 1) shell are most likely to be absorbed. The energy rule is the same as that for 0- decay: If the resulting atom is lighter than the original atom, it is unstable to K capture. The changes resulting from various nuclear processes are often represented in a proton-neutron diagram (Fig. 17.4) in which each nucleus is plotted in terms of the number (Z) of its protons versus the number (A — Z) of its neutrons. It is a result, of the processes we have just discussed that no two adjacent isobars (nuclei with same mass number) can both be stable. The heavier will #-decay into the lighter. The energies of electrons and positrons from decay have been determined with various types of /3-ray spectrometers. In principle, they measure the momentum of an electron by (hiding the curvature of its path in a known magnetic field. It is found that electrons in a given type of 8 decay may have any energy up 248 Looking In: Atomic and Nuclear Physics &T, ~ Orig. nucleus K CQpt , A-Z Rg. 17.4 A proton-neutron diagram. to the calculated energy release Q (Fig. 17.5), Here is a difficult; with the hypothesis that & decay consists of the emission of an electron (or positron) and the conversion of a neutron to a proton (or proton to a neutron). Tor the nuclear change is from one state of definite energy to another state of definite energy. Yet the electrons emitted carry varying amounts of energy, up to the maximum available. There is another difficulty. Consider the # decay of a nucleus containing an even number of nucleons. Its angular-momentum quantum number is an integer, since there is an even number of spin-i particles present. If a single electron is Fig. 17.5 A continuous spectrum. now created, there will be an odd number of spin-£ particles and the total angular-momentum quantum number will be half an odd integer. But a spontaneous change in angular momentum is not possible. To remove these difficulties, we assume that along with the electron, another particle, also of spin4, is created and emitted, Radioactivity 249 but not observed! This particle is called the neutrino. Since it shares the disintegration energy Q with the electron, the con- tinuous energy distribution observed for the ff particles (Fig. 17.5) can be explained. The neutron is assumed to have zero rest mass, so the only change needed in our previous equations is to replace E k by E k + SfcneutrUo- The neutrino participates only in reac- tions. Since it has no rest mass, it travels with the speed of light. It is postulated to have spin ■$ and to obey Pauli's exclusion principle. The neutrino lias no electric charge, and it is difficult to detect! This remarkable particle has been assumed as necessary by physicists since about 1934. Its existence was first experi- mentally demonstrated in 1956, by detection of y rays produced in a planned sequence of events initiated by the neutrino. 17.8 NATURAL RADIOACTIVE SERIES In experiments which followed the discovery of radioactivity, quite a number of substances were found to show activity. It was found that certain of these substances were associated with each other in series, the successive members being formed by the dis- integration of the preceding member, until a stable nucleus is reached. One can predict that there should exist four separate decay chains or radioactive series. A nucleus belongs to one of four classes, depending on whether its mass number A has the form 4n, 4n -+- 1, 4» + 2, or 4n + 3, where n is an integer. Radioactive decay of a nucleus in one of these will result in the formation of daughter nuclei in the same class. This follows since there is no change in mass number in decay or in 7 decay, while in a decay, A.l = 4. The four radioactive series are represented in Fig. 17.6. Each bears the name of its longest-lived ele- ment. The neptunium series is not observed naturally because gaNp ! " (T = 2.2 X 10 s year) has almost completely decayed since tin- formation of the elements (about X 10' years tigo). The decay schemes of these four series end with stable isotopes of lead. A few radioactive isotopes which do not belong to the heavy-element chains are found in nature, Table 17.1. When the elements in a radioactive series are allowed to accumulate, a steady state will be reached (if the parent atom has a long half-life) in which the number Nx\i of atoms of one isotope 250 Looking In: Atomic and Nuclear Physics N-A-Z Thorium series (A = 4n) ? -z, >Th" J - 140 Ra Ht ( V A Lrv."* Th 14 'Ho /• ' 130 Ph r; Tl Pb !0S N = A-Z 140 Neptunium series (A-4H + I) Np J " 130 \ \ y\ 1 R Li»«' Jf Ml r J"V /■A Pc s " Pb*"^. r" Tl™ 1 N=A-Z 80 84 88 93 Z Uranium series (A = An + 2) yJM SO 84 88 92 Z Actinium series (A = 4n + 3) -6 u ; j» 140 Th iJ ' 131 Ac'V 1 * r itoJ Th' 37 C 111 Rr,*" 130 I s " Pc- 15 Pb ,u / 1 . X 11 Hb 80 84 88 92 Z 80 84 88 92 Z Fig. 17.6 Decoy schemes of the four families of natural radioactivity. which decay per unit time is equal to the number A/oX* of atoms of the* next isotope which decay per unit time, or JV.Xi = N 2 \2 - iVaXa = ■ ■ ■ (equilibrium} (17.21) This equilibrium equation is often used to calculate X for an Radioactivity 251 Table 17,1 Isolated natural radioisotopes Isotope Decay Half-life (years) isotope whose half-life is too large or too small to make a particle- counting experiment convenient. PROBLEMS 1. Radium E has a half-life of 5.0 days. Radium E emits a 0- partiele to become radium P. (a) Which nucleus (E or F) has the greater positive charge? (b) Starting with 1.0 gm of radium E, how long would it take for -J gm to decay into radium F? 2. a particles shot vertically upward arc deflected by the earth's magnetic field in which direction? 3. Calculate the mass of Au l,s (7* = 2.7 days) in a source of 1.0 mc. Ans. 4.0!) X 10-" gm 4. Five mg of IV '" {T - 140 days) are allowed to decay for 1.0 year. What is the activity of the sample at Hie end of that time? Ahx. 1.35 X 10" disintegrations per second 5. A sample of radioactive sodium (Xa- 1 . T = 14.8 hr) is assayed at 95 mc. It is administered to a patient 48 hr later. What is the activity at that time? Ans. 10 mc 6. What is the volume of 1.0 mc of radon. M Hn m (T - 3.82 days), at 0°C and I atm pressure? Ans. fi.fi X 10~" m ! 7. Suggest a method for using data on the uranium-decay series to estimate the age of the earth, Suggest B waj of using the radioactive isotope of carbon C u (T = 5,600 years) to substantiate the age of cot- ton fabrics found in an Egyptian tomb. iH 3 r 12.4 j Created continuously > by cosmic radiation .C" r 5,590 J in atmosphere nK«« §,K 1.2 X 10" ^Rb" r 6.2 X 10"> ».lo ,li r 6 X 10 14 stLo"" r 2 X 10" M Sm 147 a 1.5 X 10" 7ito 176 r 2.4 X 10'° 76 Re"< r 4 X 10 12 18 Nuclear Reactions No man will ever comprehend the real secret of the difference between the ancient world and our present time, unless he has learned to see the difference which the late develop- ment of physical science has made between the thought of this day and the thought of that. T. H. Huxley A particle directed at a nucleus may undergo a collision (elastic scattering) which leaves the struck nucleus unaffected. A second possibility is that a nuclear reaction takes place producing sonic change in the struck nucleus. The incident particle may be absorbed into the struck nucleus. A rearrangement may occur in which the incident particle remains in the nucleus and another particle emerges. The incident particle may emerge but leave the nucleus in a different energy state. There are other possibilities. Nuclear reactions may be caused by individual nucleons, photons, deuterons, a particles, and heavier particles. 252 Nuclear Reactions 253 The first artificial nuclear transformation was achieved by Rutherford in 1919, in bombarding nitrogen with a particles from a natural radioactive source, Ha C Because of the impor- tance of neutrons in nuclear reactions, we shall depart from historic sequence to discuss first the discovery of the neutron by Chadwick in 1932. Among the achievements of nuclear studies are the production of scores of valuable isotopes, the discovery of the neutron and other particles, and the release of energy in the processes of nuclear lis.sion and fusion. 18.1 DISCOVERY OF THE NEUTRON Bothe (1930) found that when a particles from polonium fell on a beryllium foil, a penetrating radiation was emitted. Irene and Frederic Joliot observed (1931) that the intensity of this radiation was apparently increased by passage through paraffin. They sug- gested that Bothe's radiation was y radiation which knocked out fast protons from paraffin and other hydrogen-rich substances. Chadwick (1932) applied the equations for the Compton effect to the head-on collision of the assumed y ray and proton (mass m) and showed that the maximum energy given to the proton by a photon (hv) would be 2hv/(2 +- mc*/hv). Experi- mentally the recoil protons from paraffin were found to have a maximum energy of 5.7 Mev, requiring thai the y ray from Be have energy hv ")."> Mev. Hut when aifcrogen was subotitvted for paraffin as a target, the i.'2-Mev recoil nitrogen ions which were observed required that the same y ray have an energy of 90 Mev. Chadwick resolved this contradiction by suggesting that the "rays" from Be were actually neutrons, whose existence had been proposed by Rutherford in his mode! of the nuclear atom. The fact that atomic masses (beyond i\V) are roughly twice the atomic number suggests that the two types of particle neutron and proton which constitute a nucleus have approxi- mately equal mass. Chadwick confirmed this expectation by calculations made on the reaction 5 B" + a IIe<-> ,»' + 7 N M + Q 08.1) Three of the four masses were known. The energy of the incoming a particle (from Po) was known. The value of Q was determined 254 Looking In: Atomic and Nuclear Physics Nuclear Reactions 255 from the observed increase in kinetic energy. The mass of tlic neutron was thus found to ho 1,00(57 amu. 18.2 NUCLEAR FORCES; STABILITY OF NUCLEI The hypothesis thai atomic nuclei sire composed of neutrons and protons is now well established, and the term "nucleons" is used to refer to these nuclear particles collectively. The size of the nucleus is estimated by bombarding atoms with high-enerLiy electrons and counting Imw many of them score direct hits. The radius of a nucleus containing .1 nucleons is found to be approximately Iin, n = 1.2 X ID »,!' m (is. 2- An atom is stable because of the Coulomb force of attraction which binds the electrons to the nucleus. Within the nucleus, however, the Coulomb forces exerted by the protons are forces <>]' repulsion which tend to make the nucleus unstable. The emission of a particles from nuclei and nuclear fission (Chap. 20) are evidence of this. Somehow the repulsive Coulomb forces within a nucleus must be counterbalanced by strong attractive forces, different from electrical and gravitational forces. The nature of these nuclear forces is only partly understood. We shall discuss some of the facts which arc known about nuclear forces. An important, distinctive property of unclear forces is (heir short range. The nuclear force between two nucleons becomes negligible if they are separated by more than about 1,4 X 10 -16 m. In contrast, gravitational and electrical forces have no upper limit on the distances over which they may act. A second property of nucleus forces may lie deduced from a graph of the binding energy per nucleoli /•;,, .1 against the number of nucleons A (Fig. IS. I). Kxccpt for the lightest nuclei. E B .1 is approximately constant, about 8 Mev per micleon. Thus the total binding energy increases approximately in proportion to the number of nucleons in the nucleus: K„ a A. (The relation for a Coulomb force would be B« = A 2 .) This relation implies that a given nucleoli is bound not to every other nucleoli present, but only to its nearest neighbors. Then the addition of more nucleons increases the total binding energy only by an amount proportional to the number of nucleons added; K H /A does not change appreciably. Present evidence indicates that the nuclear force between two protons is the same as the force between two neutrons and that these may be equal to the force between neutron and proton. The last property of nuclear forces which we shall mention is pairing. The stable nuclei usually have even numbers of protons 10 ■ * / ■ — f- 20 40 60 80 1 00 1 20 1 40 1 60 T 80 200 220 240 A— Fig. 18.1 Binding energy per nudeon as a function of mass number A. and of neutrons (Table 18.1), Only the four light elements iH s , a Li°, sB 1 ", and 7 X U have odd numbers of both neutrons and protons, and for these elements the numbers of neutrons and protons are equal. Table 18,1 Evidence for pairing Neutron number (A — Z) Proton number Z Even Odd Even Odd 160 52 56 4 256 Looking In: Atomic and Nuclear Physics When a plot of neutron number versus proton number is made for all nuclei (Fig. 18.2), one observes a gradual increase m the neutron/proton ratio with increasing Z. This is explained by the fact that the Coulomb (repulsion) force between protons increases more rapidly as the number of protons in the nucleus increases than does the effect of the nuclear force between protons. This difference in the behavior of the Coulomb pp force and the nuclear pp force accounts for the gradual decrease in E B /A from 160 140 120 Nuclear Reactions 257 i "* 100 ■a E 80 p ^ 60 40 20 -t- « — / • / • / s / s /* £ . ■_ Fig. 18.2 Neutron -proton plot for stoble nuclei. 20 40 60 80 100 Proton number, X about 8.8 Me? for A near 50 to approximately 7.0 Mev for A = 240 (Fig, 18.1). 18.3 NUCLEAR-REACTION EQUATIONS We shall consider some possible outcomes when a particle or nucleus ,r strikes a nucleus X resulting ha the emission of particle ?/ and the obtaining of nucleus J": z + X— }■ + (18.3) The notation is often abbreviated as X(.c,y)Y, where the first symbol stands for the struck nucleus, the symbols in parentheses stand for the incoming and outgoing particles, respectively, and the symbol following the parentheses represents the residual nucleus. The reaction associated with Chadwick's discovery of the neutron, Eq. (18.1), may thus be abbreviated as Be 9 (a,n)C™. Before artificially accelerated particles became available, about 1932, only 10 nuclear reactions were known, all of the (a,p) type. It seems probable that in the majority of artificially produced unclear reactions the first step is the formation of a compound nucleus. The projectile and the target nucleus coalesce. The com- pound nucleus is unstable, because of its excess energy. It emits one or, sometimes, more particles of high energy to regain stabil- ity. When Rutherford bombarded nitrogen with a particles emitted by Ra C (li)li)), he initiated the first nuclear transmuta- tion by artificial means. The equation describing it in terms of a compound nucleus is ,He< + ,!?"-♦ [«F l *J-»«0" + ,11' (18.1) The same compound nucleus (but not in flic same energy state) could be produced by other reactions The breakup of the unstable compound nucleus usually depends only on its energy state, not directly upon the particle that pro- duced it. There are often several possibilities; for example, U*Zn 6i l -^ w&l« + T -+, Cu" + ,H J The 3 o*Zn Si may also eject other particles; ill 2 , iH*, sHe 1 or two n', but the probabilities of these reactions are low. Present nuclear theory does not permit prediction of the way a particular compound nucleus will break up. bike chemical-reaction equations, nuclear-reaction equations must be balanced. The total electric charge (the number of pro- tons) must be the same before and after the reaction. The total number of nucleons (neutrons and protons) must be the same, before and after the reaction. Together, these requirements mean that the number of neutrons must be the same before and after the reaction, likewise the number of protons. (There are two 258 Looking In: Atomic and Nuclear Physics exceptions: If we regard ft* decay as a "reaction," then since there is no incoming particle, the number of neutrons changes by ± 1 and the number of protons changes by + I. At extremely high energies, greater than 2 Hcv, it becomes possible to create micleou pairs. In such reactions, which we shall not discuss, the number of nucleoli* docs not remain constant.) 18.4 THRESHOLD ENERGY In a nuclear reaction .r 4- .Y — ► 1" 4- >j, the net increase in kinetic energy is called the disintegration energy Q. This Q is the net decrease in rest mass, expressed as its equivalent energy: Q = H + m x ) - (m v + Tn. u )]c l Q = initial rest energy — final rest energy (18.5) Since Q is the amount of rest energy eon veiled into kinetic energy, Q is often called the energy release of the nuclear reaction. For an encounter which results in elastic scattering, Q = 0, If the Q value of a reaction is positive, the reaction is called exo- thermic. Such a reaction can occur for incident particles of any kinetic energy. If Q has a negative value, the reaction is called emlolhermic. Example. Calculate Hie Q value for the read ion T X 14 + n' -» 7 ,V l * 4- y ,V - I i.odt.vji; 15.011)512 n' = 1.00898(5 X' 6 = i;-).illMs7s 16.016512 m = O.OlKiS-i amu Q = 931(0.01 KM) Mev = -H0.K Mev Conservation of momentum imposes a condition on induced nuclear reactions, as it does on all other collisions. This condition is particularly important for reactions with negative Q value. Prom energy considerations alone, one would think that ir the incident particle x approached the target nucleus (at rest) with a kinetic energy A'*.* = Q, then the reaction would occur. But then the momentum would not be conserved. The initial momentum is greater than zero, but the final kinetic energy, and thus the final momentum, would be zero. So, actually the incident particle must have enough kinetic energy B** so that the outgoing particles can have the same total momentum as the incident particle. The Nuclear Reactions 259 minimum value of A"*.., which makes the reaction possible is called the threshold energy. The minimum value of /;**. z which satisfies the equations for both conservation of energy and conservation of momentum is found to be Threshold - (£*.,) , ni n = (l + M Q (18.0) Example. Find the threshold energy for the reaction ,|jm + )H i _* ,()!* + p „i (q = _3.4 S m,. v ) Threshold = ( 1 + — ) Q = ( 1 + ^§1)3.48 Mev = 3.72 Mev \ "».v/ \ 14.00// PROBLEMS 1. State the number of protons and neutrons in each of the following nuclei: ,[,i* 6 I!e"», fi C 13 , 1S S 3B , and n Ui tm . 2. The nuclear read ion ,l.i + ,II»->2-.Hc* + Q liberates 22.4 Mev. Calculate the mas-, of JLa* in amu. (I)eutcron = 2.014180 amu, a particle = 4.00:3873 amu.) 3. Imagine that a free neutron gives off an electron and changes into a proton. Calculate the energy Q which is consumed or liberated in this process. What does your answer suggest about the stability of free neutrons? •Ins, Q = 6.79 Mev 4. When neutrons :ire produced by bombarding deuterons with dcutcrons, the reaction is represented by ,H*+ ,II i -. ! lle a + o" 1 + Q The neutrons produced in this reaction will have at least how uiurh energy? Am. 15 Mev plus the kinetic energy of the bombarding deuteron. .">. As the source of I he energy radiated by stars, it has been sug- gested that a series of nuclear reactions such as this carbon cycle occurs: C" + H' ..V + 7 Q =+l.»o Mev X la -» C 1 ' + e + + neutrino (K t $) , = 1.20 Mev C" + H' — X 14 4 7 Q ' = +7.58 Mev \u + H'^0" + 7 Q = +7.34 Mev O lb -* N" + e* + neutrino (A\ a),,,.,, = I .UN Mev K' B + II 1 -* C 12 + lb' Q = 4-4.98 Mev Write the equation which represents the net result of this whole cycle. Ans. 4H l — * He* 4- 2e v + 2 neutrinos 4- energy 19 Absorption of Radiation Science has a social value, and the man of science cannot wash his hands of his discoveries. It is his duty to see that they are used for the betterment of mankind, and not for its destruc- tion, Q Fournier To interpret experiments in nuclear physics and to apply the knowledge gained from them, it is necessary to know how the high-energy particles behave as they pass through mutter. For this discussion, high-energy particle means one whose kinetic energy is much greater than the ionization energy of the atoms or molecules of the material in which it is passing. We shall discuss the absorption of radiation chiefly in relation to the identification of particles, the measurement of radiation dost;, and the prob- lems of human health. 19.1 TYPES OF RADIATION Jn the behavior of a high-energy particle the most important fact is whether or not it carries an electric charge. A particle which 260 Absorption of Radiation 261 carries an electric charge (as do the electron, positron, proton, deuteron, and a particle) will exert a force on each electron near which it passes. A charged particle collides with many electrons in traveling even a short distance in matter. In many of those collisions, the struck electron is knocked out of its atom. The incident charged particle loses its kinetic energy as it leaves behind a trail of ion pairs (ejected electron and ionized atom). A stream of charged particles is referred to as an ionizing radiation. Photons and neutrons which carry no charge do not necessarily collide with every electron near their paths. Streams of uncharged particles are called nonionizing radiation. 19.2 DETECTORS Ionizing particles are easy to detect electrically. In an ionization chamber, a metal cylinder C has a wire II' insulated from the cylinder along the axis. The tube is filled with gas at low pressure, and a potential slightly less than that reiumed for a discharge is maintained between cylinder and wire. A thin window allows particles, say, a particles, to enter the chamber. Kach particle ionizes the gas, producing a rush of charge and a fall of potential at P which actuates a counter circuit. Thus one can count the number of a particles. The behavior of the ion pairs created can be studied by plotting a curve of the size of the current pulse versus the voltage applied to the tube. The ionization chamber Bottery ; 1 Capacitor # To amplifier and counter resistor Fig. 19,1 Ionization chamber particle counter. (Fig. 19.1), the proportional counter, and the Geiger-.Muller counter are ionization instruments designed to operate on dif- ferent regions of the curve. A scintillation counter makes use of one of several substances 262 Looking In: Atomic and Nuclear Physics which, when struck by a single particle, convert some of the energy received in the collision into visible tight. About. ]«)()•( investigators <rf radioactivity watched and counted the flashes of light which individual a particles produced in zinc sulfide. Since l!)-l I a scintillator or phosphor such as a clear crystal of naph- Al foil reflectors Photo cathode semi transparent First dynode Tenth dynode JJ "£ -".-.j— - "- Col lector grid Output Fig. 19.2 Scintillation counter. thalene has been used in conjunction with a pilot onmlfiplier, for automatic counting. A particle or a -y-ray photon entering the phosphor causes a flash of light which is reflected by the aluminum foil onto the photocallmde. Klcctrons are emitted from it, and these are subsequently multiplied to produce a relatively large pulse at the output of the tube. A cloud chamber, invented by C. T. It. Wilson in 18117, per- mits us to see the path of a particle through a gas. It consists of an enclosure filled with air and some vapor at a temperature just above, the condensing temperature. The chamber is designed so that its volume may be suddenly increased. This expansion Absorption of Radiation 263 depresses the temperature of the vapor below its "dew point." Some of the vapor will now condense. A vapor condenses prefer- entially on charged particles, as nuclei for droplets, if there are any present. So, if the gas has been traversed by a particle which ionized molecules along its path, the vapor will condense on these ions and the path of the particle will be visible as a trail of liquid droplets. Photographic plates were used by Bccquerel in his discovery of radioactivity (1886). Recently the manufacture of special emulsions for nuclear research has revived the use of this type of detector. Nuclear emulsions contain about 10 times the concen- tration of silver halide as do ordinary photographic emulsions, and are much thicker. Xuclear emulsions can be made sensitive to slow neutrons by incorporating small amounts (I per cent) of lithium or boron, which undergo an (n,a) reaction. Emulsions may be "loaded" with other elements (such as uranium) to study specific reactions. In film badges, the general darkening of the photographic emulsion, on development, measures cumulative exposure to radiation. In autoradiography, the distribution of radioactive material in a tissue or mineral section is determined by placing the specimen in contact with a photographic plate, in the dark, and developing the resulting pattern. The bubble chamber, invented by D. A. Glaser in 1952, takes advantage of the instability of superheated liquids for bubble formation, much as the Wilson cloud chamber uses the instability of supercooled vapors for droplet formation. The cloud chamber and the bubble chamber have similar general characteristics as particle detectors. The resetting time is longer than lor counters. The advantages of the bubble chamber lie in the high density (greater absorption) of its sensitive material and its ability to recycle in a few seconds. Bubble chambers filled with liquid hydrogen offer simplicity in interpreting collisions with protons, without contaminating elements. 19.3 DETECTION OF NEUTRONS A neutron is attracted to other near nucleous by the nuclear force, but it is neither attracted nor repelled by an electric charge. Since a neutron and an electron exert no forces on each other, they do not collide. (We can neglect for practical reasons the 264 Looking In: Atomic and Nuclear Physics extremely small gravitational force between an electron and a neutron and also a small electromagnetic force associated with the magnetic moments of the two particles.) Since nuclei occupy only a small fraction of the volume of matter, neutrons are pene- trating radiation, traveling relatively large distances between collisions. When a collision does take place, either the neutron is scattered or a nuclear reaction occurs. Since neutrons do not betray their presence directly in de- tectors (Sec. 19.2), they must he detected by the ionization which results from some nuclear reaction of scattering. For slow neu- trons (having kinetic energy less than I ev) it is convenient to use the reaction JB»-f oNi-» Ji»-|- ,He l If a counter tube is filled with a gas containing boron, BK 3 , or if the wall is coated with boron, then some neutrons will he captured to give fast a particles, which will cause ionizations in the gas and give counts. Another method used to detect slow neutrons makes use of the reaction on 1 + ^In' 16 -* win" 6 + y The radioactivity of an indium foil after exposure to a neutron beam is a measure of the number of neutrons which passed through the foil. The (n,y) cross section, or probability of cap- ture, is sharply higher for neutrons of l.4(S-ev energy. Thus this detector favors or picks out those neutrons. The detection of fast neutrons, and the initiation of certain important reactions, often requires first that the neutrons be slowed down. This is accomplished by arranging for the neutrons to pass into a moderator— a material such as graphite or D,,0 in which the probability (cross section) for scattering is much larger than that for a nuclear reaction. The neutrons then bounce around among the nuclei until both reach an average energy of !i/,-r, where k is the Boltzmaun constant. Bxampk. Find the energy of a "thermal neutron" in n moderator at 22 C B k = |(1.3H X 10-" joulc/K°)(295°K) • 6.11 X 10~" joule = 0,0382 ev Absorption of Radiation 265 19.4 ABSORPTION OF PHOTONS Photons can interact directly with the electrons of the material through which they pass. But for high-energy photons, the cross section (probability) of such interact ions is so small that the photons constitute an extremely penetrating radiation. The energy of photons can lie dissipated in three different kinds of collision. In the photoelectric effect a photon is absorbed by an atom; its energy is used to eject an electron and to impart kinetic energy to the electron. The cross section for the photoelectric effect increases rapidly with increasing atomic number (Z) and de- creases rapidly with increasing energy (hr) of 6he photon. In pair production, the energy of the photon is converted into a positron and an electron and their kinetic energies. The cross section for pair production increases rapidly with increasing Z of the absorber and with increasing energy of the photon, above the threshold value of 1 Mev (= 2m c 2 ). In the Campion effect, photons are in effect scattered, not absorbed. A photon is still in play after the collision. The cross section is a slowly varying function of (hv) and Z. The detection of photons is relatively simple; for any type of collision described above gives a fast electron: a photoelectron, a Compton electron, or an electron-positron pair. The electrons are ionizing particles and may be counted directly. The variation of photon "absorption" by each of these proc- esses is represented in Fig. U)M, where for each process, an absorp- tion coefficient a is defined as the product of the cross section a of the reaction and the number n of atoms per unit, volume, a — rur. If the Compton effect were strictly an absorption, a total absorp- tion coefficient a, could be defined for photon absorption Ctt — OfphutM "T" G^air I ^Complin ami the attenuation of a beam of x rays or 7 rays could be repre- sented by the exponents! equation While this relation has practical usefulness, it must be applied with care, since eeoinpton does not relate to a true absorption. 266 Looking In: Atomic and Nuclear Physics a, cm" ' |. 1.4 i I l a niol \ ^*m ^ 1 J* 1.2 ■ ^ 1 1 /? \ ' \ A 1.0 0.8 \ 1 V Jya. ra „ » x \ A 0.6 - \%~4 0.4 V s / 0,2 v^ -»-»_ 0.5 5 50 Photon energy 500 Mev Fig. 19,3 Variation of photon absorption coefficient, a, in load, with photon energy. 19.5 RANGES OF HEAVY CHARGED PARTICLES Charged particles heavier than electrons experience frequent, collisions with electrons in passing through matter. The heavier particle cannot lie appreciably deflected, and il can lose only a small fraction of its energy in collision with an electron. Vet the collisions are so frequent that charged particles are slowed down to thermal energies in very short distances. Charged particles are not a penetrating radiation. A proton with 10 Mev of kinetic Distance traveled, S Fig. 19.4 Kinetic energy vs. distance troveled for a charged particle. Absorption of Radiation 267 energy travels only 0.0 mm in aluminum; a 10- Mev a particle travels only 0.00(> mm in aluminum. The decrease in the kinetic energy of a charged parlislc with distance traveled is indicated schematically in Fig. 19.4 as occurring in many small steps. The distance traveled before the kinetic energy is all lost is called the range of the particle. Range depends on the particle, its initial energy, and the absorbing material. When the kinetic energy of the charged particle has been reduced to a small value (about 100 ev for a proton), it becomes increasingly probable that the ion will capture an elec- tron and end as a neutral atom. 19.6 ABSORPTION OF ELECTRONS AND POSITRONS The path of an electron or positron is longer than that of a heavy charged particle of the same energy, but it is a path full of bends because of scattering. Electrons, like other charged particles, lose their energy in a very small region of space; they do not constitute a penetrating radiation. 19.7 RADIATION DOSE The dose of any kind of radiation received by an object is the amount of energy that the object absorbs from the radiation. One might try to use a calorimeter to measure the energy ab- sorbed by a specimen in terms of the resulting rise in its tempera- ture. It turns out that- even a lethal dose of radiation produces an undetectable rise in the temperature of a biological specimen. Radiation produces many specific effects on physical, chem- ical, and biological systems. Many of these effects seem closely related to the ability of the ionization caused by the radiation to promote particular chemical reactions. Hence methods have been devised to specify dose in terms or ionization. A beam of x rays or y rays is said to give a dose of one roentgen (1 r)* if it will cause 2.08:{ X 10" J ionizations in 1 cm 3 of dry air at "The National Bureau of Standards Handbook H47 gives the defi- nition: "The roentgen shall be the quantity of x or y radiation such that i he associated corpuscular emission per 0,001293 gin of air produces, in air, ions carrying 1 esu (if quantity of either sign," The figure 0.001293 268 Looking In: Atomic and Nuclear Physics Absorption of Radiation 269 0°C and 1 atm. An ionization chamber is used to measure the dose from the radiation. The radiation passes through the air between the plates, and the ionization occurring in the air is collected. The chamber and its electrometer can be calibrated to read directly in roentgens. A widely accepted human tolerance dose rate is 0.3 r per week. The dose from cosmic rays at the surface of the earth is about 2 per cent of this tolerance dose. The roentgen was defined for photons. To extend the unit to permit measurement of radiation dose from other particles, and in living tissue, the roentgen equivalent physical (rep) is desig- nated as the radiation which produces the same energy as one roentgen of x- or 7-radiation. This amounts to 97 ergs per gram of tissue. This value is based on the observation that for any particle and any gas the average energy lost by a fast charged particle per ion pair formed is about 33.5 ev, A third unit for radiation dose is the red: the radiation which produces LOO ergs per gram of tissue. 19.8 BIOLOGICAL EFFECTS OF RADIATION Living tissue is damaged 1 >.V exposure to high-energy radiation. The danger is insidious, for the observed biological effects may be delayed for periods ranging from a few days to years, depend- ing upon the type of radiation and the dose received. Among the effects of overexposure to radiation are a decrease in the number of white blood cells, loss of hair, sterility, cancer, cataracts (chiefly from neutrons), and destruction of bones. In addition to the damage to the person receiving the radiation, there may be genetic effects extending through many generations of offspring. Penetrating radiations are effective in producing mutations or changes in heredity. X rays, y rays, and particles from supervoltage accelerators penetrate tissue readily and constitute externa! radiation hazards. In general, « and /S particles have low penetrating power, and Kin is the muss of I cm 3 of dry air at 0°C and 1 atm. Since 3 X 10 s esu of charge = 1 coul, I r produces 1 3 X 10» statcoul/coul 1.6 X 10-" coul/ion = 2.083 X 10» ion pairs/cm' damage from external sources will be confined to a thin layer of tissue. But a and emitters become internal hazards when intro- duced into the body in foods or otherwise. The various kinds of radiation damage seem to he statistical in nature, with no threshold or "safe" minimum exposure below which no injury occurs. Hence it seems prudent, to avoid all unnecessary radiation exposure. Since some exposure may be necessary for some people, responsible agencies have suggested tolerances, such as a whole-body exposure of 0.3 r per week when continued over a long time. I'or hands and feet the tolerance may be 1.0 r/ week. A single exposure of 25 1- in an accident can prob- ably he accepted. A whole-body exposure of about 500 r would probably be fatal, statistically, to 50 per cent of persons so exposed. 19.9 ATMOSPHERIC CONTAMINATION FROM NUCLEAR WEAPONS TESTS The probable effects on the health of the world population of atmospheric contamination arising from nuclear weapons tests cannol be assessed reliably from data known at present. Vet on the basis of incomplete information and conflicting interests, political decisions about nuclear detonations must be made which vitally affect our national defense and the freedom and health of generations to come. If one examines, in addition to research reports, some 10 official statements made since 1 !>.">(> by the Congressional Joint Committee on Atomic Energy, the United Nations Scientific Committee on the Effects of Atomic Radiation, The National Research Council, and the (British) Medical Research Council, one finds that these responsible bodies are in agreement on the following points: 1. Radiation exposure of the world population from fallout (including Si'"") as a result of tests through mid-1963 is small compared to natural background radiation and other man- made radiation (such as diagnostic x rays). 2. Any amount of radiation, however small, may carry a small but finite risk of increasing the genetic mutation rate of the population. 270 Looking In: Atomic and Nuclear Physics 3. Tt is unknown whether or not there exists! a threshold radiation dose for the production of somatic effects, including leukemia, bone cancer, and general life shortening. 4. Calculations of biospheric contamination in the event of con- tinued testing of nuclear weapons are intelligent guesses at best, since conclusions depend on the many assumptions that must be made. 5. Continued testing of nuclear weapons will increase biospheric contamination and consequent risk to the world population. Accelerated testing as more nations become nuclear powers, and (he touching off of nuclear war, could result in a serious radiation hazard to world health. 19.10 DISPOSAL OF NUCLEAR WASTES Nuclear power ranuol be developed by present techniques with- out also producing radioactive waste materials which are harmful to man. The safe disposal of such radioactive wastes is far more difficult than that of ordinary industrial wastes. More than ti;5 million gal of highly radioactive nuclear wastes are now con- fined in million-gallon underground tanks because they are too "hot" to dump. Although the concrete and steel tanks are ex- pected to last several decades, their contents will still be too radioactive to dump when the (auks have deteriorated! There has been increasing local public protest against the dumping of nuclear wastes into the oceans, relatively close to the shore ; particularly by citizens ul Cape Cod, Texas, and Mexico. It has also been pointed out that it may even be dangerous to dump nuclear wastes in remote and deep trenches of the oceans because (I) experiments increasingly indicate thai there is con- siderable circulation of ocean waters and (2) marine organisms tend to build up small and nearly harmless radioactive levels in sea water to potentially dangerous levels in the food supply. At the present time there are four general sources of radiation which can harm the present and future generations. In order of intensity, these are (1) medical and dental x rays, (2) radioactive sources naturally present in the earth, (3) radioactive fallout from nuclear testing, and (4) waste products from nuclear reactors. Within a decade or two, the latter two sources of radiation exposure may become the most important. Absorption of Radiation 271 SUGGESTED READING Articles in the Bulletin of the Atomic Scientists. The Milk We Drink, Consumer Reports, March, 1959. Fallout, in Our Milk, Consumer Reports, February, 1960. The Huge and Kver-iiiereushig Problem of Radioactive Wastes, Con- sumer Reports. February, I !)(!(). Fallout 1963 . . . an interim report, Consumer Reports, September, 19(iH. I^utgham, Wright, and B. 0. Anderson; "Biospheric Contamination from Nuclear Weapons Tests through 1968," Los Alamos Scientific Laboratory, University of California, I.os Alamos, X.Mex. 100 pp. Contain.* bibliography of 7 I ilems. I often say that when you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot express It in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be. Lord Kelvin Life would be stunted and narrow if we could feel no significance in the world around us beyond that which can be weighed and measured with the tools of the physicist or described by the metrical symbols of the mathe- matician. Sir Arthur Eddington Accurate and minute measurement seems to the non -scientific imagination a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long-continued labor in the minute sifting of numerical results. Lord Kelvin It does not take an idea so long to become "classical" in physics as it does in the arts. K. K. Darrow 20 Unconventional Energy Sources . . . the discovery with which we are dealing involves forces of a nature too dangerous to fit into any of our usual concepts. Congressional Record, T87S, commenting on the gasoline engine A physicist, like other persons, often finds living more purposeful and satisfying when he haw both short- and long-range goals. Some physicists seek to relate their goals to some of civilization's long-range problems: food production, world peace, education, and the exploitation of new sources of energy. It would seem that physics could contribute most directly in finding new sources of energy to supplant depleted reserves of coal and oil and to meet the ever increasing demand for power for industry, transporta- tion, and the home. Since we never create energy, it might be more precise to speak of a search for new and practical energy- conversion devices. Some possible sources of energy are so speculative that they are referred to as esoteric sources. The term "unconventional" is 272 Unconventional Energy Sources 273 reserved for those untapped sources ahout which enough is understood today so that one may reasonably predict that engineering refinements will soon make of them practical energy sources, important in our economy. Nuclear reactors, thermo- electric, thermionic, ami magnetobydrodynamic generators, solar cells, and fuel cells give promise of becoming increasingly impor- tant practical sources of energy. 20.1 NUCLEAR FISSION When, in 1042, the book "Applied Nuclear Physics" (K. Pollard and W. L, Davidson) was published, its title sounded visionary. Since then we have witnessed important and varied applications of nuclear physics. The nuclear reactor has heen developed into a practical source of electric power. (A reactor may become the ultimate source of power for space travel.) With particle acceler- ators and nuclear reactors, a host of new isotopes have been created. These have been important in further fundamental studies. They have also found diverse practical applications. In 1934, Fermi and his collaborators attempted to produce elements beyond the normal limit at uranium. In bombardment of the lighter elements by slow neutrons, the element after the capture is usually transformed by electron emission into the ele- ment of next higher atomic number. Therefore, one might expect that a similar bombardment of uranium (Z = 92) would produce a new element (93). This reaction has been produced with neptunium (93) as the resulting product. Neptunium also dis- integrates by emitting a (i particle to produce plutonium (94). Plutonium is a rather stable clement having a half-life of 24,400 years. From 1944 to 1950, four other new elements were produced in the cyclotron: americium (95), curium (96), berkclium (97), and californium (98). More recently elements einsteinium (99), fermium (100), mcndelevium (101), and nobelium (102) have been reported. In 1939, Halm and Strassmann found one of the products of neutron bombardment of uranium to be a radioactive barium sijBa 139 . There must then be another fragment such as 36 Kr associ- ated the barium fragment to make the charges equal. Ncir separated the isotopes of uranium in a mass spectrograph and found that «U s,b is the one that undergoes the splitting process 274 Looking In: Atomic and Nuclear Physics called fission. Fission is a new type of radioactive process, the first that produced particles more massive than a particles. In the process of fission of uranium there is a decrease in total mass, and therefore there is a corresponding gain in energy. Such a reaction then is a possible source of energy. This energy is con- trollable since the process can be started at will and its rate can be governed. Among the products of fission one finds one to three neutrons. These neutrons are faster than the ones used to start the fission, but if they strike uranium nuclei, they can cause fission. Since the fission produces the starting particles and releases energy, the reaction can perpetuate itself, provided there is enough uranium present so that the neutrons produced will hit other uranium nuclei. Thus a chain reaction can be set up. The smallest amount of material in which a chain reaction (constant neutron flux) can be set up is called the critical mass. 20.2 NUCLEAR REACTOR A nuclear reactor is a device for utilizing a chain reaction for any of several purposes : to produce power, to supply neutrons, to induce nuclear reactions, to prepare isotopes, or to make fissionable material from certain "fertile" materials. Typical components of a reactor are: the fissionable fuel (LI or Pu), the moderator (graphite or D a O to slow down the fission-producing neutrons), the control rods (usually Cd strips, whose insertion captures neutrons and slows the fission rate), and the coolant (water, air, hydrogen, or liquid metal, such as \a). In power reactors, the coolant, through a heat exchanger, may furnish steam to operate a conventional turbine and elec- trical generator. Breeder reactors make new nuclear fuel from fertile substances which cannot themselves sustain a chain reac- tion but which can be converted into fissionable material. One possible breeding reaction is iNP^^MPO** (20-1) T -, * t^l. 2'A ruin '2:.i duy» 20.3 FUSION Nuclear energy can also lie released by fusion of small nuclei into larger nuclei if in this process there is a decrease in mass. In such Unconventional Energy Sources 275 a process the two positively charged nuclei must come into con- tact even though there are strong electrical forces of repulsion. This requires thai I lie particles he moving with high speeds. With artificial accelerating apparatus, a few nuclei are given very high speeds. Only occasionally will such a particle strike another nucleus before it has lost too much of its energy to make contact. Thus the process is extremely inefficient, and more energy must be supplied to initiate the fusion process than is realized from the reaction. The necessary condition for a controlled nuclear-fusion process is the attainment of high particle energies for a time interval long enough to bring about kinetic equilibrium. Knergy must be sup- plied initially to attain temperatures about 2 X 10 7 °K (at which thermal fusion occurs in stars). At the same time reactants must be confined. Ordinary walls will not suffice, for they would vaporize under bombardment of high-energy particles, and these would be quickly cooled below their fusion temperature. These problems of heating and confinement must be solved in any con- trolled-fusion reactor. The choice of fuel for a eon t rolled-fusion reactor is made on the basis of availability and the probability of attaining with it the necessary high temperature. One would prefer elements of low atomic number because of the low Coulomb barrier to be overcome in the fusion reaction. Possible fusion reactions are shown in Fig. 20.1. Initial heating first strips the electrons from the atoms to pro- duce a "fourth state of matter," a fully ionized gas, or plasma. Further heating of the plasma is done by adding electric energy, in part by using the resistance of the plasma to produce familiar Ohmic (or Joule) heating. Suitably designed magnetic fields provide a sort of magnetic bottle to confine the ions at I0*°K. In the pinch effect, a cylin- drical current (10° amp) contracts because of electro magnet it- forces (parallel currents attract each other). The plasma inside is thus compressed, producing very high temperatures. The simple pinch is unstable, but with suitable stabilizing fields thermo- nuclear temperatures have been attained for confinement time of about 0.001 sec. Thus far, however, the power required for these devices has exceeded the useful power gained from the fusion process. THE FOUR STATES OF MATTER 1-Solid 2- Liquid 3- Gas 4-Plosrao tlili First .hres itafei of roaMer vory with arrongenien. and movement of maleculei, ihe ima'lleir parr.c1.ev C K« roc t •>■ lii ; c of a rrva i«f io I . tn to I id , mo lee u 1 ei or* cfoYe-packec 1 ond trill ■ In liquid they tnovt about within limiti. In a got, moleculei ore man (coflered Qnd movv Foiier, Fooith lfoie^ ploinna, it wholly '"ionized" 901. Molecule* break into aTorm, alonm into poiilive ion* and rega-ive elecrron>>. • Proton O Neutron -THE FUSION REACTION- Deuteriurn Fusion He3 + P Energy En ergy 3,25 Mev ■ 4 Mev Pi^rC^ Deuterium -tritium fusion He* ^^5 M^ fusion con take piece within a plasma, Fuiian ii combination of nuclei (atom* minus electrons} of certain lighl element!. Man of the Fuiion product! rl let* than that of orioinal nuclei; the difference h radiated os energy, mojlly heat. The fcjn"« voit 1 HEATING THE PLASMA ■;'-..: - Direction of current-. . •-■■.... To get controlled ihermonucleai reaction in- stead of explosion, small quonliriei of plasma must be contained and heated. Process begins with passage 0! a current through the plasma inside a Straight nr doughnut -shaped tube. energy is from hydrogen fusion . On earth most likely such reaction involves Twtopes (voiiants) of hydrogen — deuterium and tritium. In o plasma heated to millions of degrees, they may fuse, as in the H-bomb. THE PINCH EFFECT -PROBLEM OF INSTABILITY- mm A tuirenl eieaies o magnetic field around itself. This Field exerts pressure on plasma, "pinching," it toward center,, compressing St, making it hotter and preventing plasma pari ides from touching walls af tube . But this is theoretical behavior. -CONTROLLING INSTABILITY- In practice pinched column of ploimo develops "kinky," Pinch wandert llighHy; distortion af ■he magneric Held create* new forcer and diirorn column further. Pinch eilher rouchei walk of r^re rub* and loiei energy (A) or ii broken or or (B) . Powige of new current (A) around rube create! a linear meaner ic field in column/ giving il "back- bane/' Currenli induced in wo 111 of lube (B) help uraighren column. Pinch can then be mointoined longer. Fig. 20.1 Principles of a thermonuclear reaction, f Copyright by Trie New/ York Times. Reproduced with permission.) 276 Unconventional Energy Sources 277 20.4 THERMOELECTRIC CONVERSION The direct conversion of heat to electricity on a commercial scale is a prospect that has fascinated scientists and engineers for decades. In 1821, Thomas Sccbeck noted thai heal -applied to one junction of a circuit containing dissimilar metals would cause a small electric current in the connected circuit. The physical median ism can be understood, qualitatively, in terms of the free- electron picture of conduction. Kach metal contains some free electrons. These electrons can be made to move by an electric field or by a thermal field. If heat is applied at one end of the conductor, the electrons will rearrange to become somewhat more sparse in the warmer regions of the .specimen and more dense in the colder regions. This leads to an electrical gradient. To take advantage of it, the circuit is closed through a dissimilar metal (Fig. 20.2). Then, as long as the temperature difference is main- tained, the difference in electrical gradient in the two conductors will cause an electron flow, here clockwise. The efficiency of conversion, using the best metal combina- tions, was only 1 to 3 per cent. Thus, until recently the only practical application of Seebeck's effect was in thermocouples to measure temperatures. Recent discoveries in the field of semi- conductors have led to substantial improvement in thermoelectric conversion efficiency and foreshadow practical thermoelectric generators of power. One arm of the thermocouple may be made of an «-type semiconductor, in which the voltage difference is established by the flow of negatively charged electrons. The other arm may be a p-type semiconductor in which the voltage differ- ence occurs by the flow of positively charged voids (holes) vacated by the electrons. The attractiveness of materials for thermoelectric converters can be specified by a figure of merit Z defined as Z = 4 (20.2) where T = temperature, °K S = Seebeck coefficient, volt/K° (i.e., emf developed per unit temperature difference in the specimen) r/ = electrical conductivity, (ohm-cm) -1 (i.e., reciprocal of resistivity p) k = thermal conductivity, watts/ C° cm 278 Looking In: Atomic and Nuclear Physics Both 8 and a depend on the density of conduction electrons in the specimen, as shown in Fig. 20M. It is apparent that for intermediate- and low- temperature use, semiconductors will pro- vide the highest efficiency in thermoelectric converters. Pairs of semiconducting comp< ds which have high conversion effi- I mutators Semiconductors Metals Fig. 20.2 A thermocouple circuit of dissimilar metals, A and 8. The migration of electrons from regions of higher density toward regions of lower density produces o con- ventional current in the counterclockwise sense. ciencies have been found by making binary ami ternary com- pounds of materials in groups I, III, and VIj or the periodic table: AgSbSe^, CuTiSt, etc. The numerous combinations possible make the task of screening and developing the most favorable thermoelectric materials a formidable one. Vet exciting progress 10" Electron density, no./cm* Fig. 20.3 Properties that govern the choice of materials for thermoelec- tric devices, (Courtesy John C. Kelly, VVesfinghouse Research laboratories.) 40 - 30 c HI 9 a. £. 20 o c .2 'o LU 10 Practical limits of thermoelectricity Central station esel or marine Present ^^ ^S Automobile Auxiliary power J ! -L. -L. 10 100 1000 10,000 Power rating, kilowatts 100,000 Fig. 20.4 Thermoelectric power devices con be competitive with other power sources. {Courtesy John C. Kelty, Weslinghouse Research laboratories.) 279 Fig. 20.5 Power- producing thermoelectric elements mode of germonium- silicon semiconductors. (RCA laboratories, Princeton, N.J,) Heat source Nuclear Nuclear heat transFer Fossil Fuel ^"Waste" heat r Junction technology Thermoelectric materials Controls DC-motching voltage/ current inverters Fig. 20.6 Thermoelectric power system alternatives. {Adapted from the Genera/ Electric brochure GEZ-3Q79B.} 280 Unconventional Energy Sources 281 has been made. Seebeck's original thermocouples (1821) could convert heat into electric power with an efficiency of only 2 per cent. Study of the PbS-ZnSb couple by Maria Telkes in 1833 raised the efficiency to 4 per cent. Further work with semi- conductors has given the present efficiency of about 17 per cent. Theoretical considerations (using quantum mechanics) suggest that it will be possible to attain efficiencies as high as 35 per cent. This will make thermoelectric power devices competitive with existing power sources (fig. 20.4). Each of the germanium-silicon thermoelectric elements shown in Fig. 20.5 is capable of gener- ating about -i watts upon exposure to heat at about 1000°C. A s<]uare-foot platelike arrangement of snch elements could generate up to 10 kilowatts, nearly three times the usual electric power demand in a home. Some alternatives to be explored in the development of a thermoelectric power system are suggested in Fig. 20.0. 20.5 THERMIONIC CONVERTER Thermionic emission was noticed by Edison in 1883. In 1956 V. C. Wilson designed a converter in which electrons are "boiled J^Xv. Cooling Insulator 1 "*- ia — 1 — Anode f Electrons | MM* Cathode Load KS Heat Fig. 20,7 A thermionic converter. out" of a hot metal and used to produce an electric current directly. One obvious difference between the thermionic con- verter (Fig. 20.7) and the thermocouple is that in Wilson's device the metals arc separated by a vacuum or a gas at low pressure. There is electrical Row between the electrodes, but there 282 Looking In: Atomic and Nuclear Physics is less flow of heat in this space than through a metal. Thus t ho electrodes can be at different temperatures, and the efficiency is increased. The conversion process is shown in l'"ig. 20.8, where electron energy is plotted against distance from cathode to anode. The base line corresponds to the energy of the electrons in the cathode. Heating the cathode "lifts" some of these electrons over the work- function barrier at. the surface of the cathode, w n into the space between electrodes. If the electrons can follow path a to the anode with only a small loss of energy, there will be a potential rs I V I \ I \ b I V \ ~~ «^- - < Cathode > 140Q°K Fermi level 77777777777777^ .1 a gas ^ Plasma drop | T Anode ~ 700° J" Fermi level ' Load — r Output voltage Fig. 20.8 A plot of electron energy vs. distance (cothode to anode) in a thermionic converter. difference between the electrodes, capable of doing work in an external circuit. In vacuum devices, the electrons entering the interelectrode space soon form a space-charge barrier, represented by path h. This would increase the cathode electron energy neces- sary to electrons to cross to the anode, so the space charge is neutralized by adding an ionizable gas, such as cesium. Or alternatively a vacuum-type converter is made with a very small {0.001 in.) spacing between cathode and anode to minimize space- charge effects. Current models of thermionic converters are stated by Gear eral Electric to have these characteristics: vacuum type, efficiency 5 per cent, cathode temperature 1100°C; gas-filled type, effi- ciency 17 per cent, cathode temperature 15:!0 °C. The gas-lillcd Unconventional Energy Sources 283 unit has the additional advantage of smaller weight per unit of power: -! versus 2."> li> kilowatt. 20.6 MAGNETOHYDRODYNAMICS An Mill.) generator utilizes the principle discovered by l-'araday that an ion moving in a magnetic held experiences a side push (Sees. 10.8 and 10.9), Hot ionized gas is forced between the poles of an electromagnet (Fig. 20.9), producing a voltage difference To regenerator Hot gos Flow Flow ©- Field Current JV -VW Fig, 20.9 A magnetohydrodynamic generator, between the electrodes, at right angles to the magnet. By con- necting the elect rudes, power may be delivered to an external load. A regenerator is used to recover energy from the emerging Kas stream which may still be as hot as 2000°C when its ionization has dropped to levels insufficient for effective energy conversion. An MUD generator might be operated as part of a conven- tional gas or combined gas and steam turbine cycle. Few data exist today on which to calculate efficiencies attainable with such a combination . Some estimates suggest that addit ion of an M H D 284 Looking In: Atomic and Nuclear Physics generator could raise tlio over-all efficiency of a generating station to 55 per cent. 20.7 FUEL CELLS A fuel cell is a continuous-feed electrochemical device in which the chemical energy of reaction of a fuel and air (oxygen) is con- verted directly and usefully into electrical energy. A fuel cell differs from a battery in that (1) its electrolyte remains un- changed and (2) it can operate continuously as long as an externa! supply of fuel and air is available. Sir William CSrove, an Englishman distinguished in electro- chemistry and the law, used a hydrogen fuel cell in his experi- Chemieol energy r * Heat \ *■ Thermoelectric -*■ Thermionic * *■ Thermogolvanic Fuel cell / Fig. 20.10 Fuel cells convert chemical energy directly into electric energy, thereby avoiding the thermodynamic limitation on the efficiency of heot engines. ments as early as 1839. By the end of the last century, Wilhelm Ostwald and others came to appreciate, through thermodynamic analysis, that the fuel cell is potentially the most efficient simple way of converting chemical energy into electrical energy. Heat engines are subject to the Carnot limitation of thermo- dynamics which says that the maximum theoretical efficiency with which heat can be converted into another form of energy is determined by the inlet and exhaust temperatures of the engine: 7> 71 J i nlet / outlet , on rt\ Maximum efficiency = 7',,, i,-i It is an attractive feature of the fuel cell that its efficiency is not subject to the Carnot limitation, for the energy being converted never deteriorates into the random motion of heat. The fuel cell, when compared with familiar methods of generating electric 1 4) w I a.'-* \ f -« 0. 1 + o x 2 £ -2 I « .2 -£ K V ■5 2 t S3 $MWiW AW Mi '" '' IV* 1 'if V 1 '.I: 'r 1 lit. M j ',] : 1 \\Ai\i I. 1 ^F± 'v — ^ ,1 I I, I 1 p 1 1 "< III ('"ill i! f 'if jf 1 li 1! . MMii , hi 1. if-. llil'.i,li.l.l,r l |l.,.il)|l:M l l,'„lt,ll.r l t'^L 91 c «l CD >- K o p o. o 285 286 Looking In : Atomic and Nuclear Physics Fig. 20.12 A 75-wott 4-cell Allis-Chalmers fuel cell system designed for and tested under "jsero gravity" conditions. energy (Fig. 20.10), is very direct in its conversion of chem- ical energy into electrical energy. Partly because of this incentive, fuel cells are probably the most highly developed of the uncon- ventional energy-conversion methods discussed in this chapter. Under favorable conditions, efficiencies of 80 and even 5)0 per cent have been reported with hydrogen fuel. A fuel cell, like any other electrochemical cell, contains two electrodes: anode and cathode. These are joined externally by a metallic circuit, through which the valence electrons from the fuel flow, and internally by an electrolyte, through which ions flow to complete the circuit (Fig. 20.1 1). These are the electrode reactions : Anode 2H 2 - 411+ = ee~ Cathode Oj 4- 4H++ ier = 2H 4 Over-all O* + 211, = 2H a O Unconventional Energy Sources 287 The electron does useful work for its in passing from anode to cathode in the external circuit. The hydrogen ion completes the circuit by going from anode to cathode through the electrolyte. The electrons are urged through the external circuit by the thermodynamic driving force called the Ciibbs free energy of the over-all reaction. The major difficulty noted by Grove in 18W) is still a problem in the design of fuel cells: how to obtain sufficient fuel-electrode (catalyst)-eleetrolyte reaction sites in a given volume. In many Fig. 20.13 Unlike other conver- sion systems, fuel cells ore more efficient at tow output. Fuel cell 50 100 Rated bod, per cent cells, fuel (gas), electrolyte (liquid), and electrode (solid) are brought into effective contact by a porous electrode structure which depends on surface tension forces to get reasonable contact stability. In theory a fuel cell can be built in almost any size and capacity. Practically, fuel cells are packaged in small modules or "batteries" to be connected in series or parallel as needed for it particular application (Fig. 20.12). While conventional gener- ating devices hei ie less efficient as they so front design load to idling, the fuel cell is more efficient at lighter loads (Fig. 20.13). APPENDIX Reaction Thrust The concept of reaction Hi nisi may lie clarified by considering the recoil produced by a parallel si ream of particles. Prom Newton's laws it follows that for any system <>f objects or particles tin- center of mass of the sy-iem moves according to the equation F = , mv at (1) where F = net external force applied to system m = total mass of system v = velocity vector of the center of mass / = time Xo matter how complicated the system or how inncli force one of its parts exerts on any other, if the net external force is zero (as in field- free space), then It™ il (2) which stales thai (he total moment urn of Hie system is a constant vector i|ii:inl ity. Consider a system of two particles, a "rocket" of mass m ami velocity and a particle of (jas of mass 8m which is just leaving the nozzle with rcla- live velocity t.. The uel momentum of this system is mv + 8m{r — v,). From Eq. (2) j- [mv + 6m (v — iv)] = 289 290 Appendix A Rut 8m(di</dt) is negligible, and d(6m)/dt = -ilm ill. since t he mass of exhaust gas equals the decrease of the mikei max. Also for the exhaust velocity r, . ilr, ill = 0, and m is a small quantity which approaches sera in the limit. We have the result ilr dm m — = — v. dl dt (3) or lit (4) where F is the reaction force on the rocket. The mass flow leaving the rocket dm/<lt is re presented by a positive Dumber. The negative sign in the equation expresses the fact that Fand '•, are in opposite directions. APPENDIX B Burnout Velocity and Range In differentia] notation, Kq. (:i.bs) of A><-. :i7 may be written du rfflll m) (i) Kven if the thrust is not constant-, this equation can be integrated to give the velocity !>, at burnout «i — i'o = ffu/, In h gtt, cos 8 flit (2) If we assume that the rocket starts from rest. c„ = 0, set /. = i.\. n "g, and R = Wo/»i&, the ratio of initial mass t" final or burnout mass. Kq, (2} may be put in the form i'» = '"-tr In R — gtt. eos 9 (3) Here 6, is the duration of burning in seconds. The two averages f, (( and ;/ are necessary since the values of both effective exhaust velocity and gravitational acceleration are dependent on altitude. The altitude reached at burnout for a rocket in drag-free vertical flight with practically constant thrust (dttt/dt = const) turns out to be h = g„I,k f 1 - p _ . 1 ~ il?«'i + <Vb + Ita W 291 292 Appendix B where Ao is the initial altitude at the start of burning. After burnout, the rocket will coast upward to its maximum hoi^Iii h m , Again assuming vertical nielli and negligible (has. hut taking into account the variation of g with altitude during coasting, (lie masting distance is A c = >;■ (r. + ft*)' 2{?d r, 5 - t-»*(r, + h b )/2g tt («) where r r is the radius of the earth. For a rocket which reaches a summit of no more than a few hundred miles, h, is much smaller than r, ami \±i\. (5) rednees to a familiar form lu M (6) The summit altitude A„ reached in this vertical flight is h m = A* + A c ^j To approximate the range <>f a ballistic rocket, one may treat the powered portion of the trajectory as vertical ami the coasting portion a< elliptic. The coasting range a, along the surface of a nonrotating earth has been determined as - 2r f sin- 1 rf 2g„r, -** (8) The range calculation can be corrected for the earth's rotation by using for i>i the vector sum of relative burnout velocity and the velocity of the launching site and by adding veclorially to h l the distance the land- ing point move's while the rocket is in flight. If %» small, Bq. (8) reduces to i he familiar equation for the range of an ideal parabolic trajectory- on a Hat earth: 0u (9) If r b is large, but less than (2ff r ( )i, t he denominator of Fq. (S) approaches zero and s r becomes r f . Hence a burnout velocitv of (2ffur,.)J is just sufficient for the rocket to enter into a circular satellite orbit. The optimum angle of elevation * »f the trajectory at burnout vanes with the desired range * r according to the relation tan * - 1 - sin (s r /2r r ) ooa (sJ2r,) For short ranges, + = 46", For longer ranges, * is Jess than t.V (10) APPENDIX Schrodinger Wave Equation If * is the amplitude of the de ftroglic wave, W W ** £ m The X in this equation is to be found from the momentum of the particles we are discussing. The momentum /' can be related to the kinetic energy I mV Ei _ ' ^ . JL lir p = V^B, in 2m <2) The total energy B of a particle is its kinetic energy K, : plus its potential energy E„, so h a ~ p~ V2m"c¥^re7) and the wave equation becomes aso, x-iy ^ity JU-1 dx- dy* ^^ 1 A* (3) (4) E. Schrodinger showed, in I92(i, that Bohr's rules of quantisation could be explained on the basis ( ,f the solutions of this equation. The quantity ty (jisi) is called the "wave function" or the "probability amplitude." Although + may be negative (or even complex), it turns out that its 293 294 Appendix C We may ask what sort of eolations this equation would have for foe electrons moving m the +x direction. Sinee no forces are applied to theeteetrons, they move with constant velocity. Their potential energy W the same at all pointe; we may take /' equal to zero. The solution of he wave equation in tins ease will be a plane wave, expressive in terms of Bmea and cosmos, just as for an electromagnetic wave. APPENDIX 1. BOOKS FOR A PHYSICS TEACHER'S REFERENCE SHELF American Association for the Advancement of Science: "The Traveling Ilijdi School Science Library." AAAS and National Science Founda- tion, Washington, D.C., 1961. American Institute of Physics: "Physics in Your High School," McGraw- Hill Book Company, Inc.. New York, I960. Hrown, Thomas II. (edL): "The Taylor Manual of Advanced Under- graduate Experiments in Physics," Addison- Wesley Publishing Company, Inc., Heading, Mass., 1959. Deason, 11. J. (ed.): "A Guide to Science Beading," The New American Library of World Literature, Inc.. New York, 1963. Glasstone, Samuel: "Sourcebook on Atomic Energy," D. Van Nostrand Company, Inc. Princeton, N..L, 195K. Hodgman. CI), (ed.): "Handbook of Chemistry and Physics," Chemi- cal Uubber Publishing Company, Inc., Cleveland, 1963. [(niton, Gerald, and 1). 11. I >. Poller: ■■Foundations of Modern Physical Science," Addison- Wesley Publishing Company, Inc., Heading, Mass.. ]!)oS. Miehols. W, C. (ed.): 'The International Dictionary of Physics ami Electronics," I). Van Nostrand Company, Inc., Princeton, N..L, 1956. National Science Teachers Association: "New Developments in High School Science Teaching." Washington, D.C.. I960. I neludes 9-page list, "Additional science program materials available." Orear, Jay: "Fundamental Physics," John Wiley & Sons, Inc.,, Now York^ 1961. Parke, N. G.: "Guide to the Literature of Mathematics and Physics," Dover Publications, lue., New York, 1958. 295 296 Appendix D Physical Science Study Committee: "Fhyaes," D. C. Heath and Company, Boston, !%(). : "Laboratory Guide for Physics," J). C. Heath and Company Boston, IWiO. Price. Derek John deSollu: ■'.Science since Babylon," Yale University Press, New Haven. Conn,, 1961. Kesuick, R., and I). Halliday: "Physics for Students of Science and Engineering," John Wiley & Sons. Inc., New York. I960. Rogers, Eric M,: "Physics for the Inquiring Mind: The Methods, Nature and Philosophy of Physical Science." Princeton University Press. Princeton, N.J., HltiO. Rouse. L J., and R. J. ISarllc: "Experiments for Modern Schools," John Murray (Publishers), Ltd.. London, 195(5. Weber, R. I.., M. W. White. :iud K. V. Manning: "College Phvsies.'* MeCraw-Hil] Hook Company. Inc. New York, 1959, White. M. W., K. V. Manning, and R. L. Weber, "Practical Physics," McC raw-Hill Book Company, Inc., New York, 1955. Includes 33 experiments. 2. SOME PERIODICALS FOR A SCHOOL SCIENCE LIBRARY American Journal of Physics. American Institute of Plivsies. 335 Last 45 St., New York 17. N.Y. Xature, Macmillan & Co. Ltd., St. Martin's St., London, WC 2, England, and St .Martin's Press, inc., 103 Park Ave., New York 17, The Physic* Teacher. American Institute of Phvsies, 335 East 45 St. New York 17, X.Y. Physics Today, American Institute of Physics, 335 East 45 St., New York 17, X.Y. The School Science Renew, The Science Master's Association, 52 Bate- man St., Cambridge, England. Science, American Association for the Advancement of Science, 1515 Massachusetts Ave., XW, Washington 5, IXC. The Science Teacher, Journal of the National Science Teachers Associ- ation, 1201 16 St., XW, Washington (i, D.C. Scientific American, (15 Madison Ave.. New York 17. N.Y. Sky and Telescone, Sky Publishing Co.. Harvard College Observatory, Cam I) ridge 3M, Mass. 3. SOME PROFESSIONAL ORGANIZATIONS OF INTEREST TO THE PHYSICS TEACHER American Association of Physics Teachers. American Institute. Physics. 335 East 45 St., New York 17. N.I . Am e rican Chemical Society. 1155 16 St., XW, Washington 25, D.C. American Meteorological Society, 3 Joy St.. Boston S, Mass. of Appendix D 297 American Backet Society. 500 Fifth Ave,. Xew York 36, N.Y, (Ask for latest Book List.) American Society for Engineering Education, W. L. Collins, National Secretary, University of Illinois, Frbana. 111. Astronomical League, 310 Livingston Terr.. BE, Washington 20, D.C. Commission on Mathematics, College Entrance Examination Board, 425 West 1 17 St., New York 27, N.Y. Committee on School Mathematics. University of Illinois. Urbana, 111, Educational Testing Service, Princeton, N.J. (The Cooperative Test Division publishes a loose leaf binder, 805 pp., of Questions and Problems in Science, Text Item Folio no. 1. 195ft.) National Association of Biology Teachers, Paul Webster. Secretary- Treasurer, Bryan City Schools. Bryan, Ohio. National Education Association, 1201 Hi St., NW. Washington (i, D.C. National Science Teachers Association, 1201 16 St., NW, Washington ft, D.C. School Mathematics Study Group, Drawer 2502A, Yale Station, New Haven. Conn. Science Master's Association, John Murray (Publishers), Ltd., 50 Albemarle St., London, Wl, England. Smithsonian Inst it uf ion, Washington 25, D.C. 4. SOME SUPPLIERS OF PHYSICS APPARATUS FOR TEACHING Central Scientific Division, Cenco Instruments Corp., 1700 Irving Park Road, Chicago 13. III., and (it Hi Telegraph Rd.. Los Angeles 22, Calif. The Ealing Corp., 33 University Rd., Cambridge 3S, Mass. Macalaster Bicknell Co., 243 Broadway at Windsor St., Cambridge, Mass. (Suppliers of PSSC apparatus.) Science Materials Center. 5!* [-'mirth Ave., Xew York 3, N.Y. The W. M. Welch Scientific Co.. 1515 Sedwick St.. Chicago 10, III. 5. GREEK ALPHABET A a alpha N V nu Ii fi beta £ xi r 7 gamma omieron a a delta ii TC l'i E e cpsilott P P rho Z r seta V a sigma H n eta T r tau n 1 liela T V upsilon 1 I iota * <P phi K K kappa X X chi A X lambda * * psi M n inn Q w omega 298 Appendix D 6. SYMBOLS = means equal to ■ means is defined as, or is identical to ^ means is not equal to = means varies as, or is proportional to 2 means the sum of * means average value of j: « means is approximately equal to > means is greater than "(» means much greater than) < means less than (« means much less than) " Vof-lln™ {hiir , ° V, ' P <!ifii,) '*&**** fiwt doubtful digit; e.g. 12,600 ik stated to only three signifieant figures: 1.20 X W 7. BRIEF INSTRUCTIONS ON USE OF A SLIDE RULE* irtwL£?a&' ^P^H? 11 » «««*»% performed on the "C" ? I »i « S T caes -,? he " um! " ,r J " on the left end of the seale is ,, t |, ( | Su£%T££S! ^"l-attherigluemlofthescalei! flufe /or multiplication: Set index of "C" seale over cither „f th« second fac or on the "C" scale. Read the answer on the •<!>" J.. under the ha,rh„e. Determine the location of the decimal point I, "i rough mental approximation. ' ■ kxamplk: Multiply 17 X 23. See Kg. D.I. (lit) Set Left "C" Index Over V on "D" Scole fo-r 1 1 ■ ■> -Tr Fig. D.l Life?.t,v3 — -a: i: :^r ^i t ..'.,i i ^i.: > ..;'' | ' , i| ii i ^ i >: t i Vt* ii i i Hbt -_. (2nd) Under 23 on "C" Scole Read 391 on "D" Scale //ow (o Dtafe. Division j s generally performed on the "C" and "D" scales also. " the^^^utS'', 8 -'' ^T 'T 1 ""' ° W tte » um ^tor (dividend) on tin- » scale and bring the denominator (divisor) on the "C" scale gen*Co. StnU ' tbnS "'"' ' ,,|)VriK!l1 iHu8 *«*ti«M <-"">!esy of Eugene Diets- Appendix D 299 under the hairline. Head the answer on the "D" seale, under the index of the "C" scale. Determine the decimal point by rough mental approxi- mation. EXAMPLE: Divide by 3. See Fig. D.2. <hr) Set 3 or." C" Over 6 on "D" r 2 c i : rt ' *? T -V h'K 1 i- i ■ u 1 1 1 iJi i T li I' * lT (2nd) Under Left "C" Index Read 2 on "D" Fig. D.2 flow to Find a 8quctre (too'.. Problems involving square roots are, worked on the "A" and "B" scales in conjmietion with the "C" and "I)'' scales. Note llial the "A" and "li" scales are divided into I wo identical parts, which will he referred to as "A-left" and "A-fight." Rule for square roots: If the number is greater than unity, and has an odd number of figures before the decimal point, set the hairline over the number on "A-left" and read the square root under the hairline on the "D" scale. If the number has an even number of figures before the decimal point, use "A-right" instead of "A-left." Locate the decimal point in the answer by mental approximation. If the number is less than unity, move the decimal point an even num- ber of places to the ri^ht until a number between I ami 100 is obtained. Find the square root of the number thus obtained as explained above. To locate the decimal point, move it to the left one-half as many places as it was originally moved to the right. kxamclk: Find the square root of 507. See Fig. D.3. (lsl) Set Hairline to 567 on Left Ho If of "A" ^ .c i.„.V4.r-.Vi" ".?■%■'■& o i 7i ■; ~i h I tTwWWl TL ii|iiii jm i|i i ii | iii|iii t} i»|i nji| i ; i liK > } i : 1 1 1 it i t f 1 1 1 : [ w ; J i w jhii | hii|h 4 hii 1 i ^ ■ » i),. - i l " \s~rs (2nd) Under Hairline on "D" Read 238 Fig. D.3 300 Appendix D l*sc "A4aft," since than is mi odd number of figures before the decimal point. By mental approximation, locate the decimal poinl after the second significant figure, making the answer 23.8. kxamplk: Find the square root of 0.0956. See Fig, ]>.-]. (1st) Set Hairline Over 9.56 on Left Half of "A" * '■■•■'•■-.' ...r W..I., .t,„. > . t . > D I .1 ■ .1 ■ .1- n n'~a •> n'TT) ~ ~m • • < ■ "■!" ),...i...*..i...j . « p j , t.i.ip ,k • O im].|.l,,.Jl^,l,l|l. ( .. , 1( , ,ii m i; ■ (2nd) Under Hairline on "D" Read 3.09 Fig. D.4 .Move the decimal point two places to the right, thus obtaining 9.56. Fse the "A-left," hecause there is now an odd number of figures before the decimal point. Take the square root of 9,r>6. !hen move the decimal poiul one place to the left, making the answer 0.309, 7A. SLIDE RULE BIBLIOGRAPHY Bishop, C. ('.: "Slide Rule- How to Fse It." Barnes & Xohle, Inc.. New York. Bshbaeh, 0. \V., and H. L Thompson: "Vector Type Log Log Slide Rule." .Manual no. 1725, llugene Dietisgen Co., 1009 Vine S( Philadelphia 7. Pa. Naming. M, L:"A Teaching Guide for Slide Rule Instruction," Pieketl and Eckel Inc., 1109 South Fremonl Ave,, AJhamhis, Calif Harold, Don: "Slide Rule? May I Help . . . ," KeulTel and Baser Co., Adams and Third Sis., Hohoken, X.J. "Inirnducing the Slide Rule," Wabash Instrument & Specialties Company, [tic. Wabash, lud., 1943. "Ii'- Easy to Use Four Post Slide Rule." Educational Director, Freder- ick Post Co., 3050 North Avondale Ave., Chicago, 111. (A projec- tion slide rule is expected to be available soon for classroom use.) Johnson. L. H.: "The Slide Rule," I). Van Xostrand Company. Inc., Princeton. X..1. -Macliovina. P. K.: "A Manual for the Slide Rule," McGraw-Hill Book Company, foe, 330 Weal 12 St.. New Ym-k 30, N.Y., 1950. "Mathematics, .Mechanics, and Physio." Engineers Council for Pro- fessional Development, 29 West 39 St., New York, N.Y. 8. TRIGONOMETRIC FUNCTIONS Radians [;•■:.]'.•.■'. Sine* Cosine? Tang ants Cotangents .0000 a .0000 1.0000 ,0000 s 00 1 . 5708 .0178 i .0175 ,9988 .0175 57.20 80 1 . 5533 .(Will 2 .0349 .9984 .0340 28 . 04 88 1 ,5359 0524 if .0523 .9986 .0521 10,08 87 1.5184 ,0698 4 .0008 .9976 .0699 14.30 86 1 5010 .0878 5 .0872 .9962 .0875 11.430 85 1 . 1835 Ml 17 . 1046 .9946 .1051 n 61 i 84 1 . (661 . 1222 7 1219 . 05)25 . 1228 8.144 83 1.4480 . 1390 8 .1392 mod3 .1 105 7.115 S2 1 .4312 .l, r >7] 9 . 1504 .9877 . 1581 0.314 81 1.4137 .1745 10 .1736 .9848 . 1703 5.671 SO I 3963 .1920 11 ions .9810 .1914 5 ! 15 79 1.3788 .2091 12 ,21)70 .978! .2120 4.705 78 [ .3614 2209 13 .2250 .9741 .2300 4.332 77 1.3439 21 !3 14 .2410 .9703 .2493 I Oil 70 1.3205 .2(il8 15 2588 0059 .2070 3.732 75 1.3090 .2798 16 •J 7511 .9613 .2867 3.487 74 I .2918 . 2967 17 .2021 , 9563 .3057 3.271 73 1.2741 .3142 18 .3090 .9511 .3240 3.078 72 1 .2666 ,3310 19 .3250 .0455 ,3443 2 904 71 1.2302 .3491 20 .3420 .9397 .3640 2.748 70 1,2217 , 3605 21 3584 .9336 .3839 2.606 69 1.2043 . 38 H> 22 .3746 9272 .4040 2 175 08 1.1808 . 101 1 23 .3907 '.1205 . 4245 2.350 67 1.1694 .-1189 21 .4007 .9136 .4152 2.246 «a 1.1519 . 4363 25 .4220 .9003 .4003 2 144 05 1.1345 .1538 211 .4384 .8088 ,4877 2.050 < 1 1.1170 .1712 27 . 4540 .8010 . 5095 1 . 963 68 1 .0990 .4887 2S 1695 .8829 .5317 1.881 02 1.0821 .5081 29 IMS .8740 .5543 1.804 01 1. O047 5236 30 .5000 . 8660 .5771 1.732 00 1 .1)172 .5411 31 5150 S572 .0009 1.664 59 1.0297 .5586 32 .5299 S1S0 0249 1 .000 58 1.0123 .5700 33 .5440 .8387 .0494 1 54(1 57 0.9948 . 5934 34 5592 .8200 . 0745 1 is:; 56 0.9774 Jill)!) 35 .5736 .8102 7002 1.428 55 0.9599 .6283 : j ,r, .5878 SOOt) 7266 1 370 54 (1 1)125 .0458 37 .0018 .7980 . 7530 1.327 53 0.9250 .6632 38 11157 7SS0 .7813 1 . 280 52 0.9076 .0807 39 .112!):! .7771 .8098 1.235 51 o 8901 .0081 40 lltiS . 7660 . 839 1 1 . 192 50 0.8727 .715(1 II .0501 .7547 8693 1 . 150 49 0.8552 . 7330 42 .0091 .7431 .9004 I. Ill 48 0.8378 . 7605 43 .0820 . 73 1 1 ,9325 1.072 47 0,8203 . 7079 44 .0047 . 7 1 93 .0057 1.030 46 0.8029 . 78, r )4 45 .7071 .7071 1 0000 1.000 45 0.7854 Coiinei Sinfti Colon genii Tongont? Degree* Rudiufis. 301 302 Appendix D 9. LOGARITHMS TO THE BASE s These two pages give the rial lira! (hy- iierbohe, or Napierian) logarithms of tnnii- bera between I and m, correcl in four places. Moving the decimal point » places to the right (or left) fa tfie number is equivalent to adding ,, limes 2,:<02(i (or n tunes 3.6974) to the logarithm. 1 '2 3 •1 5 6 i 8 2.8026 i 6052 6 B078 8.2103 11.6129 13.8165 IS 1181 18.4207 20.7233 I 2 3 4 G 6 7 8 9 1.0 i.i i.j i. a 1.4 l.S 1.6 1.7 l.S 1.9 2.0 2.1 J.J 2.3 J.4 J.S 2.0 J.7 2.8 2.6 3.0 3.1 3.2 J.S 3.4 3.5 3.0 J.7 3.8 S.9 M 4.1 4.2 4.3 4.4 4.5 4.0 4.7 4,6 4 LI 0.00OQ 0953 1823 1624 3305 4055 470(1 5300 5S7.V 0419 0.8931 7419 7885 8329 8755 9ISJ 9555 0.9933 1.0296 0647 0100 0108 iiair, 0392 1044 1133 1222 1310 IMS J070 2151 2770 2852 2927 mm 2700 1314 1632 10811 •>•>$<■ 2528 1809 3083 3350 JO 10 1.3883 4110 436! !>,, 4810 5041 6201 5476 5688 6892 1436 3607 3577 3640 4121 4187 4253 4318 4762 4824 4836 4947 53S5 5423 6481 SS39 5933 5988 8043 6098 8471 0523 6675 6627 698t 7031 7080 71JS 7467 7514 7561 7608 7930 797S 8020 8065 S372 8416 8450 850J 8700 8838 8S70 8020 9203 0243 92*2 9322 9504 9632 0670 9708 9960/0006 0043 0080 033J 0367 0403 0438 0682 0710 0750 0784 1019 1053 1086 1119 1346 1378 1410 I44J 1063 1094 1725 1756 1009 2000 2030 2060 2267 2206 2320 2355 2556 2585 2613 2641 2137 2S«5 2M)2 2M| 3110 3137 3164 3191 3376 3403 3429 3455 3635 3661 3680 3712 3888 3013 3938 3962 4134 4159 4183 4J07 4375 4398 4422 4446 4600 4633 1666 4670 4839 4861 4884 4907 5063 5085 5107 5120 5282 5304 5320 6347 5497 5518 5530 5560 5707 5728 5748 5769 6013 5033 5953 6974 0488 0583 0077 0770 0862 1398 1484 1570 1055 1740 2231 2311 2390 2469 JS46 3001 3075 3148 3221 3293 3710 3784 3853 3920 3088 43S3 5008 .i,W, 6152 6678 7178 7665 8109 8644 8901 9361 0746 0111, 0473 IIS IS 1161 1474 1787 2000 J3S4 2669 2947 3216 3481 3737 3987 4281 4469 4702 4929 51SI 6309 5581 5790 5994 4447 4S11 4574 4837 5068 5IJ8 SI8S 5J47 56.53 5710 5766 5822 6208 0259 0313 0366 0729 0780 6831 6SS1 7227 7275 7324 7372 7701 7747 7793 7839 8154 8198 8242 8286 8587 8829 8671 8713 0002 9042 9083 9123 9400 9439 9478 gsi7 9783 9821 98S8 9806 0152 0188 0225 0260 0508 0543 0578 0513 0952 0886 0918 0953 1184 1217 1249 1282 1506 1537 L500 1600 1817 1B48 1878 1009 21 19 2149 2170 JJ08 24 13 J442 2470 2499 MM 2976 3244 3507 3702 4012 4255 4403 47J5 4!>. r ,l 5173 6390 5602 S8I0 6014 2720 2754 2792 2002 3020 3056 3271 3297 3324 3633 3558 3584 3788 3813 3838 4036 4001 4085 4279 4303 4327 4510 4540 4563 474s 4770 471)3 4074 4090 5019 5195 5217 5230 5412 5433 5464 5023 5844 5865 5831 5851 5872 8034 8054 8074 19 0.0053 1823 2024 3365 4055 4700 5306 5878 6419 0.6931 7419 7885 8320 8755 9103 9565 0.9g33 1.0296 0647 1.09S6 1314 1632 1939 2238 2526 2800 3083 3350 3610 1 3m;;i 4110 435) 4586 4816 6041 5261 5476 5680 6802 1.6094 6974-3 '.HUH ."i 05)22-7 7897-10 4871-12 I Sir, I I 8819-17 6793-19 27(17-21 Tflnrhf of the Tabular Difference 12 3 4 5 10 10 29 38 48 17 20 35 44 8 15 24 32 40 7 15 22 30 37 7 14 Jl 28 34 13 19 26 32 8 12 18 24 30 8 II 17 23 J8 5 II 18 22 27 5 10 15 21 26 5 10 15 20 24 5 9 14 19 23 4 9 13 IS 22 4 9 13 17 21 4 8 12 16 20 8 II 10 20 8 II 15 19 7 11 15 18 7 It 14 18 7 10 14 17 7 10 13 18 6 10 13 16 6 II 15 6 9 12 15 6 9 12 14 8 II 14 8 II 14 8 II 13 8 10 13 8 10 13 2 5 7 10 12 2 5 7 10 12 2 5 7 II 2 5 7 9 11 2 4 7 9 11 J 4 7 9 II 2 4 6 9 II J 4 8 8 11 J 4 8 10 2 4 6 8 10 Appendix D 303 LOG, (BASE e = 2.718284) Tenrht of the Tabular Difference 9 1 I 1 4 5 6 7 8 9 10 1234 5 5.0 1.8094 0)14 6134 0154 0174 6194 0214 6233 6263 0273 6292 2 4 6 8 10 G.1 112! 12 8311 6332 6351 6371 6390 8409 8429 8448 6487 6487 2 4 6 8 10 8.2 6487 6508 66J5 6544 6563 8582 6601 6620 6630 6668 6677 2 4 6 8 10 S.J 6677 0696 0715 6734 0752 6771 6700 6808 6327 6845 6864 2 4 6 7 6.4 6864 8882 8901 6910 6938 MM 6974 8993 7011 7029 7047 14 6 7 9 6.5 7047 7060 7084 7102 7120 7138 7156 7174 7192 7210 7128 24 57 6.6 7228 7246 7J63 7281 7199 7317 7334 7351 7370 7387 7405 2 4 5 7 9 6.7 7406 7422 7440 "457 7475 7492 7609 7527 7544 7561 7579 2 3 5 7 9 6.8 7679 7690 7813 7630 7647 7864 7681 7099 7716 7733 7760 2 3 5 7 9 5.9 7760 7760 7783 7800 7817 7834 7851 7867 7884 7901 1.7618 2 3 6 7 8 CD 1.7016 7934 7951 7967 7984 8001 8017 8034 8050 8060 8083 2 3 6 7 8 6.1 8083 8090 8116 6132 8148 8185 8181 6197 82)3 ,j,,l 8245 2 3 5 7 8 6.2 6245 8262 8178 8294 8310 8326 8342 8358 8374 8390 -41')-, 2 3 5 8 6.3 8405 8421 8437 8453 8460 S4S", 8500 8516 6532 8647 8563 2 3 5 8 S 6.4 8563 8579 8594 8810 6025 8041 silSft K67J v ( ;s: 8703 8718 13 5 6 8 6.5 8718 8733 8749 8784 8779 8795 8810 8825 8840 MM 8871 2 3 5 S 6.6 8S71 8886 8901 8918 8931 -8940 8961 8976 8991 9008 0011 2 3 5 6 8 0.7 9021 9036 0051 '.Win; iw-ii 0095 9110 9125 9140 9155 9109 13 4 7 6.8 0109 9184 9100 9213 9228 9142 9257 9272 9280 9301 9315 13 4 6 7 6.9 0315 0330 9344 9369 9373 93S7 9402 9410 9430 9445 1.9459 13 4 7 T.O 1.9459 0473 948S 9502 0516 0530 9544 9560 9573 9587 9601 13 4 6 7 7.1 96UI 0615 0629 0643 0857 0671 0686 9600 9713 9727 0741 13 4 7 7.1 9741 9755 0769 emg »th 0810 0824 0838 9851 9385 1 ',-7'.' 13 4 8 7 7.3 1J870 9802 9900 9920 9933 9947 998) 9974 0B88/0O0I 2.0015 13 4 5 7 7.4 2.0015 0028 004! 0055 0069 0082 0000 0109 0122 OI30 0149 13 4 5 7 7.5 0149 0102 0176 (1I.1SI B202 0215 0229 0241 0255 0268 0281 13 4 5 7 T.fl 0281 0295 0308 0311 0334 0347 0360 0373 0386 0399 0412 13 4 5 7 7.7 0412 0425 043K 0461 0464 0477 0490 0503 0516 0528 0541 13 4 5 1 7.8 0541 0554 0567 0580 0592 0805 0618 0831 0643 0656 0660 13 4 5 6 7.9 IlW'.i 0681 0604 0707 071!l 0732 0744 0757 O70II 0762 2.07114 13 4 6 6 1.0 2.0704 0807 0810 0832 0844 0867 OS00 0882 0894 0906 ll'.llll 12 4 5 6 8.1 0010 0031 0843 0058 096s 0080 0002 1005 1017 1020 1041 12 4 5 6 S.2 1041 10S4 1080 1078 1090 1101 1)14 1126 1138 1150 1163 12 4 8 6 8.3 1163 1175 1187 1199 1211 1223 1235 1247 1158 1270 1282 12 4 5 5 8.4 1282 1204 1300 1318 1330 1342 1353 1305 1377 13SH 1401 12 4 6 6 8.S 1401 1412 1424 1436 1448 1469 1471 1483 1494 1506 1618 12 4 5 6 8.6 1518 1529 1541 1562 1564 1576 1587 1590 1610 1822 1623 12 3 5 6 8.7 1833 1645 1660 1668 16711 1601 1702 1713 1726 1730 1748 12 3 5 6 J .8 1748 1759 1770 1781 1793 1804 1815 1827 I83S 1849 1861 12 3 5 6 8.9 1861 1872 1883 1894 1906 1917 1028 1039 1050 1961 2.1972 12 3 4 8 J.O 2.1971 1083 1904 2006 1017 2028 2039 2050 2061 2072 2083 12 3 4 6 9.1 2083 2094 2105 2116 2127 1138 2148 2159 2170 2181 2102 12 3 4 5 1.2 1192 2103 2214 2225 2235 2240 2257 2288 2270 2 2 Mi 2300 12 3 4 5 9.3 2300 2311 2322 2332 2343 2354 2364 2376 2386 2396 1407 12 3 4 5 9.4 2407 2418 2428 2439 2450 2460 2471 1481 2492 2502 2513 12 3 4 5 9.5 2513 2523 2534 2544 1.555 2565 2576 2586 2597 2607 2616 1 2 3 4 F 9.8 2'Us 2628 2638 3649 21159 2670 2680 209O 2701 2711 2721 12 3 4 6 S.7 2721 2732 2742 2752 2702 2773 2783 2703 2803 2814 2v>4 12 3 4 5 0.8 2824 IS34 2844 2854 2865 2875 2886 2895 2906 1915 2 '.■:•.'. 1 2 3 4 S 9.9 2925 2935 1948 2966 1966 2978 2980 2996 3006 3016 2.3026 12 3 4 5 304 Appendix D 10. VALUES OF PHYSICAL CONSTANTS As experimental data improve, "best values" of the physical constants are recomputed by statistical methods. See, for example, K, It, Cohen, J. \V. M. Do Mond. 'I". \V. Lay Ion, and J. S. Hollelt. "Analysis of Variance of the 1052. Data on the Atomic Constants and a New Adjust- ment, 1885" Review of Modern i'hysv-s, 27:303 380 (1955). The values listed below have been rounded off from those liste<l in the paper cited and have been expressed in inks units. The physical scale is used for all constants involving atomic 0)88868. Avogadro's number: A'.i = 6.0249 X 10 10 molecules kmole Gas constant per mote: R« = 8,31 7 joules/(kmoie)(°K) Standard volume of a perfect gas: V a = 22.420 m 3 atm. kmole Standard atmosphere: i>» = 1.013 x 10* newtons/m* Speed of light in free space; c = 2.9979 X 10" m/sec Electronic charge: e = 1.0021 X 10- |!l coul Planck's constant: A = 6.6252 X SO" 3 " joule-sec Faraday constant: F = 9.652 X 10 7 coul/ kmole Charge/mass ratio for electron: e/m = 1.758" X 10" coul/kg Rest mass of electron: m = 9.1083 X 10~ 31 kg First. Uohr radius: «„ = 5.2917 X 10 ll m Compton wavelength of electron: X = h/mc = 24.203 X 10 13 m Boltamann's constant: A.- = 1.3804 x 10 S3 joule, °K = 8.617 X10 B cv/"K .'•lass-energy conversions: I kg = 5.610 X 10 s * Mev 1 electron mass 0,51098 Mev 1 proton mass = 938.21 .Mev 1 amu = 931.14 Mev 1 neutron mass = 939.51 Mev Energy conversion factor: I ev = 1.6021 x 10 "joule Rest masses: electron m = 9.1083 X 10 31 kg = ft.4870 X 10 • amu proton »t„ = 1 .0724 X 10"" kg neutron »i„ = 1.6747 X 10 - " kg Proton mass electron mass ratio = 1,830.12 (iravitational constant: = 6.67 X 10 ll newton-mVkg* Appendix D 305 11. CONVERSIONS OF ELECTRICAL UNITS Quantity Symbol Practical unit, mfcs Cgs-esu equiv. Cgs-emu equiv. Energy Current Electronic potential Electronic field Magnetic flux Magnetic induct. Permittivity of free w ( V £ a B 1 jouJe 1 ampere 1 volt 1 volt/m 1 weber 1 weber /m 2 BBS X 10" 11 coulV newton -ni ! 1.257 X 10" 6 newton/ omp- 10" ergs 3 X 10 9 jtotomp J X 10 -s statvolt 10~ 4 iv/cm statcoulomb dyne cm 5 1 1 9- 10" V*wso 10" ergs 0.1 abomp 10* ob volts 1 f ' abv/cm 10^ maxwells 10' gauss 1 1 space Permeability of free space 9 ■ 10" V«M> unll pole dyne cm 5 Note: / — = C Index Absorption coefficient, 265 Acceleration cine to gravity, 41, 43 Acoustic waves, 1(11, 103 (See a/so Wave) Activity, 240 specific. 242 Adams, ('. C, 7. 8 Actlu i- theory, 139, 1st; Ampere, 110 Ampere's law, 131, 150, 161 Amplitude, !l(i, 106 Angle of ascending node, 51 Angle of inclination, 51 Angular momentum, 55 Anode, 140 Apogee, BJ Apparatus, suppliers of, 297 Argument of perigee, 53 Wending node, angle of, 51 Asteroids, 10, 13 Astronautics, bibliography of, 7 careers in, 7 lilms on, 7 history of, 3 Astronomical unit, 9 Atmosphere, entry of, 72 Atom, 7!i earlv concept of. 78 models or, 117. 203 205. 23\ 25:1 radius of, K5 speed of, 86 Atomic Knergy Commission, 32 Atomic muss, 70, 153 Atomic mass unit, 70 Atomic number, 118, 183, 205, 211, 220 Atomic weight, 79 Auger effect, 175 307 Autoradiography, 203 Average life, 24 1 Avogaclro's number, 78, 144 Baker, K. 11.. IS, 19 Bnlmcr Series, 208, 212, 214 Hand theory of conduct inn. _M-"> Harkla, C. G, 281 Barrier, energy, 220, 244 Bertie, R. J., 290 Bauer. 0. A.., IS Beats, 197 BecquereJL H. A., 77, 203 Benson, O. ()., 7 Bernoulli's theorem, I'll Beta decay, 210 Bel si spectrum, continuous. 24S Binding energy, 299, 242. 251 Bishop, C. ('.. 390 Blaekbody, 168 Blackbody radiation, 166 energy distribution in. lit) (See also Radiation) Blunchard. C. II.. 153 Bohr. N.. 1 18, 293 Bohr atom model, 293 204,297.222 (See iitao Atom) Hook list. 205 Bore, 08, 9!) Hot he. W.. 253 Bragg diffraction, electron analogy of, 21 S Breeder reactor, 271 Breillat rahluug, 17li Brown, T. IS., 295 Bubble chamber, 203 Budierer, A. II.. 198 Bnchheiin, It. YV., 7, 70 308 Index Burnett, C. It., 163 Burnout .speed, rocket, 20, 201 {See alxo Rocket ) Cajori, !■'., 64 Calorie, ss Careers in astronautics, 7 [See also Astronautics) Curncit efficiency. 32. 2x4 Cathode, 140 * Cathode ray tube, 143 Cal lioiie rays, 1 30 Cavity radiation, L89 Center of mass, 20 ^ rotation about, 212 Chad wick. J., 253 Chain reaction, 274 Charge, of an electron, 145 Charge /mass ratio, 127. 142 Charging, 117 Chromosphere, II Circular orbit, 46, 53 Clock paradox, 195 Cloud chamber, 263 Cohen, ]■:. H., :«)4 Collisions, molecular, 86 nuclear, 252 Comets, 11), i:j Common ell erf, 176, 265 Condon, E. U., 210 Conduction of elect rieity in a tas 139, 140 Conductivity, band theory of, 235 electrical and thermal. 233 of metals, 146 quantum theory of, 234 Conic orbits, 65 Conic sections, 64 Conservation of energy, 200, 244 Constants, physical, :mm Corona, 1 1 Correspuiidence principle, 183, 214 J osmic rays, II, 14, 208 Coulomb. I It) Coulomb barrier. 220. 244 Coulomb's law, lis, 200 Cou titer, 201 (leiger-MuUer, 261 scintillation, 202 Crew, II., mi < 'fit teal mass, 27 1 Cross product, vector, 56 Curie, 242 Current, conventional, 127 direction of, 12s electric, (27 electron, 127 induced, 136 in magnet ie field, 130 in metals, Hti, 235 Daltnn, J.. 77. 7.s Damped wave, 106 Darwin, C, R. 115 Davidson, U'. ].., 273 Davisson ami Germer cxia-riim-nt 2I.S Deason, H. J., 205 De BrogUe, I,., 216 De Broglie wave, 170, 222, 230 Decai constant, 211, 25<J Deflection of charged particles, 125, Degree of freedom, S8 I 'elector, radiation, 2til Do ^'auvenargues, Marquis, 138 Dewey, John, 40 Dill ruction, 107, ISO of electrons, 2 IS of neutrons. 210 of photons. 2 IS Diffraction grating, 108 Dilat atio n of lime, I it Dirac, P. ,\. M., 216 Direction rules, for induced emf, 136 for magnetic force, 134 for magnetic induction, 134 Disintegration energy, 243 I Kspucement, 06 Disraeli, Benjamin, lis Distances, to planets. HI to stars, IS Dobie, .1. [''rank. ti7 Dopplcr effect, 112, 107 transverse, 1 14 Dose, radiation, 267 Dunne and Hunt law, 170 DnClaux, lit) Dulongand IViit law, I4S DuMoml.,1. U\ M., :«)4 Duncan, .1. c. mi Dyne. 21 Earth satellite, fit Eccentricity, .",1 Eddington, A. 8., 115. 271 Edison, T. A„ 2X1 Effective exhaust velocity. 25, 30 Efficiency of heal engine', 32. 284 Einstein, A., !t, 67, 77, 173, 175, 1X4, 104, 196 Einstein's mass-energy equation, 199 Einstein's photoelectric equation, 173 ESectiio current, 127 in magnetic Meld, 130 in metals, 148, 235 Electric field intensity, [21 Electric potential, 122 Electrification, 1 17 Electrolysis, I It Electromagnetic wave, 103, 161 energy of, (66 gamma ray, 242 plane, 103. (66 speed of. 104 x-ray, 175 Electron, 10s. 127, 130, 144 charge or, 145 and electrolysis, 111 e/m ratio for, 142 energy data for. Hi!) free, 146, 232 in nucleus. 246 shells, 227, 230 Electron configuration, 228, 230 llecfroti (iilTraetion, 2 Is Electron (low, 127 Electron theory of conduction, 146 Section volt, 124 Hectrostatic units, 1 19 demerits, periodic table of, 228 Elements of an orhil, 51 Jliot. C, 166 jllipse. 61, 64 Emerson, It. W., 77, 1S3 imf, induced, 135 Energy, binding. 20(1, 254 conservation of. 21 Ml, 244 disintegration, 243 distribution in blacklmdy radia- tion, 170 in electromagnetic wave, 105 oquipnrtilion of, SS forbidden, 237 in nuclear reactions, 243, 25S'. 273. 270 potential, 44 quantization of, 2(H), 220 of satellite. 1 1 sources of, 272 tiergv barrier. 220, 244 uerg'y levels, 2IIX. 21(1, 235. 242 qua! ion of state. 81 quilihrium, radioactive, 250 quipnrthinn of energy, 8S scape speed, 31, 45 shlmch, O. W., 301) xelusion principle, 227 Ixlinust velocity. efTeclive, 25, 30 Exponential law of decay, 230 Falling body, 42 Faraday, ML 136, 144, 150, 283 Faraday, 144 Paradays law of induct ion. 158, Kit I Fermi, K., 273 Fermi distribution, 234 Index 309 field, electric, 121 deflection of particles bv, 125, Ml gravitational, 42 magnetic, 125 deflection of particles by, 125, 120, 141 Field intensity, electric, 121 Field strength, magnetic, 105 Film lists, 7, IS. 37 Fission, nuclear, 273 Fitzgerald-Lorents contraction, 104 Flux, electric, 157 magnetic, 135 Forbidden energies, 237 Force, gravitational, 41 magnetic. 134, 142 on a current, 120 nuclear, 206, 254 Fourier series, 07, 104 Founder. (I,, 201) Franklin, YV. S., 138 Free electrons, 140. 232 Free fall. 43. 4li Freedom, degree of, 88 Frequency. 114 Fuel cell, 284 Fusion, unclear. 274 a, gravitational acceleration, 41, 43 (7, gravitational constant, 41 Galaxies, 17 Camilla decay, 242 Gas, fully ionized. 275 ideal, 80 kinet ic I henry of, S! Gas constant, so Gas discharge tube, 140, 141 i las law, so Gauss, 120 Gauss's law, 157. 150 ( lav-Iaissac, .1.. 7s Geiger. H., 203 (ieiger-Mnller counler, 261 rator principle, 186 Glaser. D. A,, 263 (ilasstone. S., 205 Goddard, H. II., 4 Gram tuoieciilar volume, 70 Grating, optical, 108 Gravitation, universal, 41 Gravitational acceleration, i/, 41 on planets. HI slandard, 25 Gravitational constant, 0, 41 ( iravitat iunal field. 42 Gravitational force, 41 on planets, 10 Gravitational potential energy, 44, 47 310 Index Gravity. 41 ( Irsek alphabet, 207 Croup velocity, l(J(l Croups, orbital, 227, 230 drove, W., 284 (iuidnnco of rocket, "(I Halm, a. 273 Half-life, 241 (See (i/.to Radioactivity) Hall effect. 233 Ilalliday, I).. 2!lli Hartung. M, I„. 300 Heat engine, ellieienev of, 32 Ilcavisido, (I., 120 Heiscubcrg, W,. 2 It; Hciscnljerg a uncertainty principle, 182 Henry, J.. 135 Herald, ]>., 300 H err iik. K., 02 Hertz, (i.. 157. KM High-ejicrgv particles. 260 Hohbs. M„ 3S Hodgroan, (". 1),, 2!>f> Hull on, (i., 205 Horace. Ill I lo vie, V., I!) Hubble, !■:. P.. go Huxley. T. H., 252 Hydraulic jump, SIM Hydrogen, Hulir model of, 207 energy levels for, 208, 210 spectrum of, 207 (See alao Atom) Ideal ga« law, SO I (u puke, specific, 25, 27, 32, 33 Inclination, angle of, 51 Induced current, 135 direction of. 135 Induced end. 135 direct ton of, 138 Induction, magnetic, 128, 131 at center of loop. 132 direction rule for, 134 Faraday's law of, IjjK force due to, 134 near Straight wire, 133 Insulator, 237 Intensity. ]{I5 wave, 165 (See alto Wave) Interference. 106, 1*7 Interferometer. iMi Interplanetary travel, OR Ion. 144. Mi) Ion propulsion, 34, 3d, 71 Ionization chamber, 201 Ionization energy, 229 Ionization potential, 2in, 211 Ionizing radial ion, 2(51 iHHtojws, 140, I />:* Ives. II. i:.. Ml Ives and Stilwell experiment, li)7 James, J. N., m Jet .separation, 2!l Johnson, L. 11., 300 Joliot. 1''.. 253 Jupiter, lit, 12 A*-eapture, 217 Kelvin, Sir William Thomson, 271 Kepler's laws, 56, (HI derivation of, (i2 Kinetic theory of gases, 81 Kirchhoffs law, lli.s Kiwi engine, 32, 33 Krogdahl, W. S.. I!) l-i ert, B., St!. S7 Launching speeds, 70 (See also space) Layton. 1'. W., 304 Lens's law, 135 Life, on planets, is, 10 of radioisotope, 241 Light, speed of, |(>4, |8(i, 1JI2 Line of force, gravilat imial, 42 Loeb, L. I).. s:i Logarithms, 3112 Lorents, H. A., 232 Lorentz transformation. 193 Much number, 2s, 102 Machoviaa. !'. K.. 300 McLaughlin. !l. ».. Ml Magnetic deflection, I3ti, 141 Magnetic field strength. I(i5 Magnetic Mux, 135 Magnetic induction, I2K, 131 ill center of loop. 132 direction rule for. 131 Faraday's law of, 158 force due to, 134 near straight wire. 133 M.'ignclohydi'odyiiariiics, 34, 283 Manning, K. V.. .".Id Marconi, (!., 157 Mars, HI, 12 Marsdon. E., 203 Mas.s. atomic. 70 niolcnilar, 7'.) Mass-energy relation, HIS Mass number, I IS Mass rat io, 30 Mass spectrometer, 131, Mil Uainbridge type, 152 Dempster type, 151 Index 311 Matter, composition of. 117 four states' of. 27(i Matter wave, 170, 217, 230 Maxwell..). C. 83, 87, 88. 103, 166, lid. 1114 Miixwell-iloltzinann distribution, s:{. 87, 232 Maxwell's electromagnetic theory, 186, Mil. 107 Maxwell's equations. Mil Mean live path, 84 Mechanics, principles of, 21 Mendeleev, 1). 1., 228 Mercury, HI. II Mesons, 107 Metals, conduction in, 232 Meteorites. HI, 13 and radio waves, 13 Miehels, W. ('.. 205 Micbelson interferometer, lst> Michelson-Morlev experiment, ISS Slilky Way, 17 Miliicurie. 242 Millikan, H. A.. 145 Mills, M. M-, 38 Missilery, chronology of, 4 Model rockets, 37 manufacturers of, 37 (Seeaiso Rocket) Models of molecules, 8(1 Modern physics, 77 Mole, 79 Molecular mass, 7'.i_ Molecular volume. 70 Molecule, 70. 117 mode! of. 88 Momentum, angular, 55 quantisation of, 209, 220 in relativity theory, 198 Moon. HI, 12 Moslev's (aw, 231 Motion. Newton's laws of, 21, 43 uniformly accelerated, 22, 42 Motors, rocket. 27. 32 (See alxo Hockot | NASA (National Aeronauties and Space Administration), 7, 8, 32 National Association of Rocketry, 37 Mangle, J. E., 14, 17. 19 Neptune, 10, 12 Neutrino, 240 Neutron, 118, 274 detection of, 263 discovery of, 253 Newell. H. K.. 14, 17. HI Newton, Isaac. (HI Newton (unit), 21 Newton's law of gravitation. II Newton's laws of motion, 21, 43 Node, 51 Nonionizing radiation, 261 Nozzle, rocket, 2S, 33 (See also Hocket) Nuclear atom model, 205, 238, 253 Nuclear landing energy, 200 Nuclear emulsions, 203 Nuclear fission, 273 Nuclear force, 206, 254 Nuclear propulsion, 31, 71 Nuclear reactions, energy from, 243. 25S. 273. 270 equations for. 25ti threshold energy for, 258 Nuclear reactor, 32, 273, 274 breeder, 274 Nuclear testing, 269 Nuclear wastes, 270 Nuclei, stable, 255 Nut ■icon. 1 17, 254 Nucleus, 1 17, 230 radius of, 254 Oersted, B.C., 110 Orbit, circular, Hi, 53 electron, 227,230 for entry o[ atmosphere, 72 in hydrogen atom. 2111), 222 satellite, 47 conic, 66 elements of, 61 precession of, 57 Ordwnv. I 1 '. L. * Orear, .1., 63, 296 Oslwald, W., 281 Page, I.., S3 Pair production. 243, 266 Pairing of aueleons, 255 Parke, X. <■',., 296 Pauli exclusion principle, 221 Performance of liquid propellents, 20 Perigee, 51 argument of, 53 Period. 51. 54,04 Periodic table, 228 Periodicals, io astronautics, 8 list of, 296 Permeability, 131 Permittivity, 1 10 Phase veloeitv. 100 Photoelectric effect, 172, 266 Pholooleotron, 172,242,286 Photons, 172, 173, 179, ITS, 180,211 absorption of. 205 Photosphere, 1 1 Physical const ants, 304 Pinch effect, 270 312 Index Planck, M.. 171 I'liini-k's constant. 17-'. 173. I7(i 182 Planck's law, 170 Planets, flight between, 68 life on, 18 physical data for, 10 limes to reach, 70 Plasma, 275 Plasma propulsion, 35 Plato, 07 Poinenre, II., 07, 139, 191 Pollard, !■:.. 273 Positron, absorption of. 207 Positron -electron pair, 243, 205 Potential, electric, 122 Potential energy, gravitational, II. 47 Potential well, in gravitational Belda, 47 for hydrogen atom, 210 Pouudal. 21 Poyntnuj'a vector, 166 Pressure of radial ion, i:t Pressure thrust, 24 Price, 1). J„ 2!iti Probability, wave. |8I) Product vector, 66 Project Hover, 32 Project Sherwood, 34 Projectile motion, 43 I'ropellants. performance of, 28 Propulsion, ion, 34, 3(1 nuclear, 31 :, solar, 36 Proton, UK, 127 Proton-neutron diagram, 243, ->|>(* front's hypothesis, 153 Q- value. 243, 25.S Quantization, energy, 208, "242 momentum, 209, 220 apace, 224 Quantum, 172, 173, 170 Quantum mechanics, 1X4. 2 Hi. 219 Quantum numbers, 224. 227 Quantum theory, iu7 Rad, 268 liailialioa. atmospheric contamina- tion by, 269 liiologieai effects of, 268 blnckhody, 108 cavity, 169 electromagnetic theory of, 156 101, 167 in space, 14 types of, 20 I Radiation holts, 14 Radiation dose, 267 Radiation pressure, 13 Hadiai ion tolerances, 17 Radioactive equuibn 250 Radioactive scries, 240 Radioactivity, decav lav, for, 239 natural. 238. 249 series in, 2-10 lypes of, 239 units of, 24 I Radioisotopes, natural, 251 Haniu, S., 74 ISAM) Corporation, 5, 7 Range, of charged particles, 266 of rockets. 2! 1. 30. 40, 201 for atmospheric entry, 73 Rationalised units, 1 10 Hay. 93, !!4 Rayleigh-Jeans Jaw, 17(1 Reaction principle', 20, 22 Reaction thrust, 2so Reactor, nuclear. 273, 274 for rocket power. 32 Reduced mass, 213 Re-entry of earth's atmosphere 72 Reference systems, 185 Relativistic Doppler effect, 114 Relativity theory, Einstein's, I 111 ma.vf ami energv in, IVx Newton's, 185 and spaee travel, 201 I win paradox in, I!I5 velocity addition in, 103 Rep, 2tl.s Resistance, of free space, 165 of metals, 147 Resistivity, 147 Resnick, *R.. 2116 Resonance, 1 10 HiKht-hiiiid rule, 134 R MS speed. 83 Roberta, Michael, 1K3 Hocket, burnout velocity for, 20 definition of, 20 flight theory of, 2 1 , 20 forces on, 29 ion, 34 mass ratio, 30 model, manufacturers of, 37 motors for. 27. 32 multiple stage, 31 oosale, 28, 33 nuclear, 31 performance of, 2ii. 36 plasma, 35 propulsion of, 21, 22, 26, 32 range of. 2IJ, 30 specific impulse of, 25, 27 staging of, 31 Rocket guidance, 70 Rocket trajectories, 60 Roentgen, w, C, 77 Roentgen, 267 Rogers, H. M., 64, 206 Holler, I). H. I)., 295 Hollett, J. S., 304 Root-in can -square speed (rtns), 83 Rouse, [..,!.. 296 Rowland, H. A., 2t)(i Russell. Hortrand, 183 Rutherford atom model, 205, 238, 2S3 Ryilbcrg constant, 207 Sarnoff. !>.. 149 Satellite orbits, 47 Satellites, earth. 3 energy of, 55 escape speed for, 31 reasons for, 0, 50 Saturn, 10, 12 .Savage, J. N„ 20 Scattering of alpha particles, 204 Schoolcy, J. S., 48 Sehrodingcr, [■',., 216 Sehrodiiigcr wave equation, 222, 220, 245, 203 Scintillation counter, 262 Screening, electron. 220 Scebeek, T., 277 Sciferl, B. §., 8,88, 58 Seismic wave, 103 Semiconductor, 237, 277 Series, radioactive, 240 Shaw, J. H., 10 Shells, electron. 227, 230 Shock wave, 102 Simultaneity, 103 Singer, ('Italics, 13N Slide rule, use of, 298 Slug, 21 Societies, professional. 296 Solar const ii at. 1 1 Solar propulsion, 30 Solar sail, 36 Solar system. 10. 18 Soli. I stale, theory ..I'. 231 Sonic boom, 102 Sound wave. 100 speed of, 101 0e» oho Wave) Space, environment of, gravitational fields in. 40, 47 radiation in, 14 vehicles in, 4 Space exploration, reasons for, li. 71 Spaee quantization, 224 Space research, 74 Space travel, and relativity, 201 Index 313 Space vehicles, chronology of, 4 Spec i lie heat, 87 of metals, 147 Specific impulse. 25. 27, 32, 33 Spectrum, I0S, 2<l(» of hydrogen. 207 x-ray. 230 Speed, burnout, 201 of light. Hi.!. ISO. I!I2 of molecules. S3, SO rms, 83 Spencer, II., 67 Spin, 225. 249 Sputniks, 5, 50, 05 Stable nuclei. 255 Stages, rocket, 31 (ate also Rocket | Standing wave, I (19, 1 1 1 Star distances, 18 State, equation of, SI Stefan-Boltsmann law, 170 Stern, ()., 86 Storn-dei huh experiment. 224 SlMwell. 0. EL III Stake's law, 145 Stouer. R. <:.. 153 Stoney, G. J., 1 44 Strassmann, F„ 273 Strughold, II., 7 Suninierfeld, M., 38 Sun. 10 Superposition principle, 104. 109 Surge. 9S Sutton. G. P., 26 Symbols, mathematical, 208 Thermion ie converter, 281 Thermocouple, 278 Thermoelectricity. 277 power from, 279 Thomas, B, It., !K3 Thompson, II. !.., 30(1 Thomson, J, J., 77, 131, 130, 149, 203. 288 Tli res hold energy, 25s Thrust, decrease with alt it talc, 24 momentum, 24 pressure. 24 rocket, 24, 289 Time dilatation. 114. 195. 201 Time to reach planets. 711 Tiros satellite. 58 Total energy of B parliele, 2(10 Transformations, Lorentz, 192 Transurauic elements, 273 Trigonometric functions, 301 Tunnel effect . 245 Twin paradox, 105 Tyndall, J., 116 314 Index I'liceifainty principle. ISI l.'nifoniily accelerated motion, 22, 42 ("nils, electrostatic, 119 tuks, 119, 120 for Newton's second law, 21 rationalized mks, I li) Utmma (film), 18 Drams, 10, 12 \'-2 rocket, 24, 30 \an Allen belts, 14, 16 Vector cross product, 50 Vector product, !2,s Velocities, iiddition of, 193 Velocity, burnout, 29 of escape, 45 group, 1 1 in of molecules, H'A, so phase, Hit) rms, 83 Venus, JO, II Volt, 122 Von Goethe, J. W., 115 Wave, acoustic, 101, 108 at a boundary, 04, 110, 245 damped, 106 in different mediums, 94 elastic, 93 electromagnetic, 93, 103, 101 energy of, 105 group velocity in, 100 intensity of, 105 in a liquid, 07 longitudinal, 92, 103 panicle motion in. 99 phase velocity in, 100 saw-tool li, 90 seismic, 103 shock, 102 sine, 96 Wave, sound, 100 speed of, 104 square, 96 standing 109, III in u string, 95, 1 11 surge, !IS transverse, 92, 103 traveling, 95 x-ray, 17.5 Wave aquation, 92, no Schroilinger. 293 Wave forms, 95, 90, 104 Wave front, 93. :it Wave mechanics, 219, 245, 293 \\.ni-|i;iniile duality, 179, 217 W ave speed, !(ll Waves, interference of, Km soperposi'i f. KM, 109 Weber, R. I.„ 7, 153,290 Weiicr, 129 Weightlessness, \i\ Wells. II. Q 00 Whipple, V. L_ 3 White, M. \V., 290 Whitehead, A. \., 50 Whitney, W. \i., 20 Whole number rule-. 153 Wiedcnianu-Frani! rule, 233 \\ nil's law. 171 Wilson, V. C, 281 Wilson cloud chamber, 262 X-ray spectrum, characteristic, 230 continuous, 230 X rays, 175 frequency limit of, 176 scattering of, J 76 ffin, II., 19 Zodiac, 9 Zwicky, h\, 08 PHYSICAL DATA, EARTH Mean diameter 12,742.46 km Angular velocity 72.9 X 10~ 6 radian/sec Mass 5.975X10" kg Mean density 5,517 gm/cm 3 Normal gravity (p = geodetic latitude) g m 9.78049(1 + 5.22884 X 10- 3 sin 3 ^ - 5.9 X lO" 6 sin 2 2<p) m/sec 2 Standard atmosphere p = 1.013 X 10 B newton/m 2 PHYSICAL CONSTANTS Na = Avogadro's number = 6.0249 X 10 M molecules/kmole R = Gas constant per mole = 8317 joules/ (kmole)(°K) k - Boltzmann constant m 1.3804 X 1Q~ 23 jou1e/°K = 8.617 X 10- 5 ev/°K y Q = Standard volume of a perfect gas = 22.420 m 3 afm/kmole c - speed of light - 2.9979 X 10 s m/sec e m Electronic charge m 1.6021 X 10~ 19 coul h » Planck's constant = 6.6252 X 10~ 34 joule-sec F = Faraday constant « 9.652 X 10 T coul/kmole Energy conversions: 1 electron volt = 1.6021 X 10- 19 joule 1 atomic mass unit = 931.14 Mev m c = Rest mass of electron - 9.1083 X 10~ 3I kg = 0.51098 Mev m p = Rest mass of proton = 1.6724 X 10"" kg - 938.21 Mev Mb - Rest mass of neutron m 1.6747 X 10-" kg - 939.51 Mev mjm s = 1836.12 e = — c 1 = Permittivity of free space = 8.8542 X 10" 12 farad/m ^o = Permeability of free space = 4tt X 10" 7 henry/m Zo = (/*oAo)i = Impedance of free space = 376.73 ohm G = Universal gravitational constant = 6.67 X 10" n newton-mVkg 2 : - : = : =-- =: -■"■■■■•■ * ■* a f* 3 fii • T 1 (0 S 68806